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Incommensurate Twisted Bilayer Graphene: emerging quasi-periodicity and stability

Ian Jauslin, Vieri Mastropietro

TL;DR

This work analyzes a lattice model of incommensurate Twisted Bilayer Graphene and demonstrates the persistence of the semimetallic Weyl phase under weak interlayer coupling, despite emergent quasi-periodicity and large-momentum Umklapp processes. The authors combine Renormalization Group methods with number-theoretic Diophantine conditions (KAM-like) to prove stability for a large-measure set of twist angles, establishing convergence of a renormalized expansion and controlling small divisors. They show that Dirac points renormalize without destroying the Weyl cone, with renormalized velocities and wavefunction factors, while confirming the framework justifies neglecting high-momentum Umklapp terms in the continuum limit. The results provide a rigorous basis for continuum TBG descriptions in this regime and lay groundwork for incorporating interactions and lattice effects in future work.

Abstract

We consider a lattice model of Twisted Bilayer Graphene (TBG). The presence of incommensurate angles produces an emerging quasi-periodicity manifesting itself in large momenta Umklapp interactions that almost connect the Dirac points. We rigorously establish the stability of the semimetallic phase via a Renormalization Group analysis combined with number theoretical properties of irrationals, similar to the ones used in Kolmogorov-Arnold-Moser (KAM) theory for the stability of invariant tori. The interlayer hopping is weak and short ranged and the angles are chosen in a large measure set. The result provides a justification, in the above regime, to the effective continuum description of TBG in which large momenta interlayer interactions are neglected.

Incommensurate Twisted Bilayer Graphene: emerging quasi-periodicity and stability

TL;DR

This work analyzes a lattice model of incommensurate Twisted Bilayer Graphene and demonstrates the persistence of the semimetallic Weyl phase under weak interlayer coupling, despite emergent quasi-periodicity and large-momentum Umklapp processes. The authors combine Renormalization Group methods with number-theoretic Diophantine conditions (KAM-like) to prove stability for a large-measure set of twist angles, establishing convergence of a renormalized expansion and controlling small divisors. They show that Dirac points renormalize without destroying the Weyl cone, with renormalized velocities and wavefunction factors, while confirming the framework justifies neglecting high-momentum Umklapp terms in the continuum limit. The results provide a rigorous basis for continuum TBG descriptions in this regime and lay groundwork for incorporating interactions and lattice effects in future work.

Abstract

We consider a lattice model of Twisted Bilayer Graphene (TBG). The presence of incommensurate angles produces an emerging quasi-periodicity manifesting itself in large momenta Umklapp interactions that almost connect the Dirac points. We rigorously establish the stability of the semimetallic phase via a Renormalization Group analysis combined with number theoretical properties of irrationals, similar to the ones used in Kolmogorov-Arnold-Moser (KAM) theory for the stability of invariant tori. The interlayer hopping is weak and short ranged and the angles are chosen in a large measure set. The result provides a justification, in the above regime, to the effective continuum description of TBG in which large momenta interlayer interactions are neglected.
Paper Structure (19 sections, 2 theorems, 142 equations, 2 figures)

This paper contains 19 sections, 2 theorems, 142 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Incommensurate TBG
  3. Perturbative expansion and small divisors
  4. Diophantine condtions
  5. Stability of the semimetallic phase
  6. The renormalized expansion
  7. Multiscale decomposition
  8. Feynman rules for the renormalized expansion
  9. Convergence of the series and persistence of the cones
  10. Beta function and Schwinger functions
  11. Conclusion
  12. Fourier transform of the interlayer hopping
  13. Proof of Lemma \ref{['lemm:dioph']}
  14. Diophantine condition from $\mathbb R$ to $\mathbb R^2$
  15. Applying the Diophantine condition to $|p_{F,i}^\omega-p_{F,j}^{\omega'}+lb+mb'|$
  16. ...and 4 more sections

Key Result

Lemma 4.1

For every interval $[\theta_0,\theta_1]\subset[0,2\pi)$, the set ${\cal D}:=\{\theta\in [\theta_0,\theta_1]$ satisfying (cond)$\}$ has measure $1-O(C_0/(\theta_1-\theta_0)^2)$, so that choosing $C_0$ small enough with respect to $\theta_1-\theta_0$, $\mathcal{D}$ has large relative measure.

Figures (2)

  • Figure 1: Example of a Feynman diagram for $S_{1,1}(\mathbf k)$ with 4 vertices.
  • Figure 2: An example of graph of order $\lambda^7$ with the associated clusters, denoted by thick rectangles. In this example, $h<h_1<h_2<h_3$.

Theorems & Definitions (3)

  • Lemma 4.1
  • Lemma B.1
  • Remark B.1