Perturbative renormalization group approach to magic-angle twisted bilayer graphene using topological heavy fermion model

TL;DR

This work develops a perturbative renormalization group framework for the topological heavy fermion model of MATBG, casting the low-energy physics as an Anderson-lattice problem with itinerant $c$-fermions and localized $f$-fermions. By deriving and solving one-loop flow equations for the hybridization $\gamma$, velocity $v_{\star}$, and Coulomb channels $U,V,W,J,J_{+}$, the authors show that under RG the ratio $U/\gamma$ generally decreases, pushing the system toward a projected-limit or Mott-semimetal regime at low energies. The flows reproduce and connect to BM model intuition, in particular the approach toward the chiral limit as interlayer couplings favor $\gamma$ growth, while $U$ remains roughly constant. The framework provides analytic insight into the interplay of topology and correlations in MATBG and is readily applicable to other moiré systems and THF-described materials, forming a foundation for constructing low-energy effective theories.

Abstract

We develop a perturbative renormalization group (RG) theory for the topological heavy fermion (THF) model, describing magic-angle twisted bilayer graphene (MATBG) as an emergent Anderson lattice. The realistic parameters place MATBG near an intermediate regime where the Hubbard interaction $U$ and the hybridization energy $γ$ are comparable, motivating the need for RG analysis. Our approach analytically tracks the flow of single-particle parameters and Coulomb interactions within an energy window below $0.1$ eV, providing implications for distinguishing between Kondo-like ($U\gg γ$) and projected-limit/Mott-semimetal ($U\ll γ$) scenarios at low energies. We show that the RG flows generically lower the ratio $U/γ$ and drive MATBG toward the chiral limit, consistent with the previous numerical study based on the Bistritzer-MacDonald model. The framework presented here also applies to other moiré systems and stoichiometric materials that admit a THF description, including magic-angle twisted trilayer graphene, twisted checkerboard model, and Lieb lattice, among others, providing a foundation for developing low-energy effective theories relevant to a broad class of topological flat-band materials.

Figures (15)

• Figure 1: Energy scales of MATBG. $v_F/a_0$ is the energy scale below which the graphene dispersion is approximated by the Dirac cone, where $a_0$ is the graphene lattice constant. $v_F/a_M$ characterizes the kinetic energy associated with the moiré lattice of period $a_M$. $\gamma$ denotes the bandgap separating the flat bands from remote bands. $U$ represents the Coulomb interaction strength between electrons in the flat bands.
• Figure 2: Band structures of the single-particle THF model fitted from the BM model with twist angle $\theta = 1.05^\circ$ and interlayer tunneling ratio $w_0/w_1 = 0.8$. The model parameters are $v_\star = 4.303$ eV·Å and $\gamma = 24.75$ meV Song:2022. Size of red dots represents the $f$-orbital weight. (a) Parameters: $M = 3.697$ meV and $v_{\smallstar} = 1.623$ eV·Å. (b) $M = v_{\smallstar} = 0$. High-energy modes above the energy cutoff $E$ (dark shaded region) are integrated out. In our perturbative RG scheme, we continue lowering $E$ until it reaches the bottom of the remote bands (light shaded region), where the running cutoff $\Lambda$ flows from $\Lambda_c$ to 0.
• Figure 3: Brillouin zone of twisted bilayer graphene. Two graphene layers are labeled by 1 (red) and 2 (blue) with a twist $\theta$. The moiré Brillouin zone (purple) is shown near the graphene $\vb*{K}^+$ valleys, with moiré lattice vectors $\tilde{\vb{g}}_2$ and $\tilde{\vb{g}}_3$.
• Figure 4: (a) $c$-fermion propagator. (b) $f$-fermion propagator. (c-d) Anomalous $cf$-propagator. The solid line represents $c$-fermion, and the dashed line represents $f$-fermion.
• Figure 5: Coulomb interactions between $c$- and $f$-fermions. $c$-fermion ($f$-fermion) is denoted as a solid (dashed) line with an arrow. Due to symmetry, $W_1=W_2$, $W_3=W_4$. $\mathcal{J}_{\alpha,\eta}^{\alpha',\eta'} = J(\eta\eta' + s_{\alpha} s_{\alpha'})/2$ and $\mathcal{J}_{+;\alpha,\eta}^{\alpha',\eta'} = -J_+(\eta\eta' - s_{\alpha} s_{\alpha'})/2$ where $s_{\alpha}=(-)^{\alpha-1}$.