Magic-Angle Twisted Bilayer Graphene as a Topological Heavy Fermion Problem

Magic-angle ($\theta =1.05°$) twisted bilayer graphene (MATBG) has shown two seemingly contradictory characters: the localization and quantum-dot-like behavior in STM experiments, and delocalization in transport experiments. We construct a model, which naturally captures the two aspects, from the Bistritzer-MacDonald (BM) model in a first principle spirit. A set of local flat-band orbitals ($f$) centered at the $AA$-stacking regions are responsible to the localization. A set of extended topological semimetallic conduction bands ($c$), which are at small energetic separation from the local orbitals, are responsible to the delocalization and transport. The topological flat bands of the BM model appear as a result of the hybridization of $f$ and $c$ electrons. This model then provides a new perspective for the strong correlation physics, which is now described as strongly correlated $f$ electrons coupled to nearly free $c$ electrons—we hence name our model as the topological heavy fermion model. Using this model, we obtain the U(4) and $\mathrm{U}(4)\times \mathrm{U}(4)$ symmetries of Refs. [1–5] as well as the correlated insulator phases and their energies. Simple rules for the ground states and their Chern numbers are derived. Moreover, features such as the large dispersion of the charge $\pm 1$ excitations [2,6,7], and the minima of the charge gap at the ${\mathrm{\Gamma }}_M$ point can now, for the first time, be understood both qualitatively and quantitatively in a simple physical picture. Our mapping opens the prospect of using heavy-fermion physics machinery to the superconducting physics of MATBG.

Figure 1

Topological heavy fermion model. (a) A sketch of the moiré unit cell of MATBG and its heavy fermion analog, where the local moments and itinerant electrons are formed by the effective $f$ orbitals at the $AA$-stacking regions and topological conduction bands ($c$), respectively. (b) The band structure of the BM model at the magic angle $\theta =1.05°$, where the moiré BZ and high symmetry momenta are illustrated in the upper inset panel. The overlaps between the Bloch states and the trial WFs are represented by the red circles. The density profile of the constructed maximally localized WFs ($f$ orbitals) is shown in the lower inset panel. (c) Bands given by the topological heavy fermion model (black lines) compared to the BM bands (blue crosses). The $c$ (blue) and $f$ bands (red) in the decoupled limit, where $\gamma =v_{\star }^{\prime }=0$, are shown in the inset. Orange dashed lines indicate the evolution of energy levels as $f\text{−}c$ coupling is turned on.

Figure 2

The self-consistent HF bands upon the ground states at the fillings $\nu =0,-1,-2,-3$. The color of the bands represent the contributing components, wherein yellow represents the $f$-electron states and blue represents the $c$-electron states.

Figure 3

The one-shot HF bands of the ground states at the fillings $\nu =0,-1$. The red solid bands are the quasiparticle bands of the decoupled Hamiltonian, where $\gamma =v_{\star }^{\prime }=J=0$. The horizontal and dispersive red bands are of the $f$ and $c$ electrons, respectively. The touching point of the dispersive red bands at ${\mathrm{\Gamma }}_M$ is quadratic, while since $M$ is small, it may look like linear. The one-shot bands can be understood as a result of hybridization between $f$ and $c$ electrons.