Origin of Magic Angles in Twisted Bilayer Graphene (1808.05250v1)

Abstract: Twisted Bilayer graphene (TBG) is known to feature isolated and relatively flat bands near charge neutrality, when tuned to special magic angles. However, different criteria for the magic angle such as the vanishing of Dirac speed, minimal bandwidth or maximal band gap to higher bands typically give different results. Here we study a modified continuum model for TBG which has an infinite sequence of magic angles $\theta$ at which, we simultaneously find that (i) the Dirac speed vanishes (ii) absolutely flat bands appear at neutrality and (iii) bandgaps to the excited bands are maximized. When parameterized in terms of $\alpha\sim 1/\theta, $ they recur with the simple periodicity of $\Delta \alpha \simeq 3/2$, which, beyond the first magic angle, differs from earlier calculations. Further, in this model we prove that the vanishing of the Dirac velocity ensures the exact flatness of the band and show that the flat band wave functions are related to doubly-periodic functions composed of ratios of theta functions. Also, using perturbation theory up to $\alpha^8$ we capture important features of the first magic angle $\theta\approx1.09^{\circ}$ ($\alpha \approx 0.586$), which precisely explains the numerical results. Finally, based on our model we discuss the prospects for observing the second magic angle in TBG.

• The paper introduces a chirally symmetric continuum model that eliminates AA coupling to mathematically predict a sequence of magic angles with perfectly flat bands.
• The analysis reveals that at the first magic angle (≈1.09°), the Dirac speed vanishes, providing a direct correlation with zero Fermi velocity and a maximized bandgap.
• Perturbative studies confirm the robustness of the first magic angle against small coupling perturbations, bridging theoretical predictions with experimental observations in TBG.

Origin of Magic Angles in Twisted Bilayer Graphene

The paper "Origin of Magic Angles in Twisted Bilayer Graphene" by Tarnopolsky, Kruchkov, and Vishwanath explores the fundamental nature of magic angles in Twisted Bilayer Graphene (TBG), focusing on isolated, flat bands near charge neutrality that emerge at these specific twist angles. The researchers offer a modified continuum model with an idealized chiral symmetry, demonstrating that at certain magic angles, the Dirac speed vanishes, absolutely flat bands form, and there is a maximized bandgap to excited states.

Main Findings

1. Continuum Model with Chiral Symmetry: The authors discuss a continuum model where the inter-layer coupling in the AA regions is negated (i.e., $w_0 = 0$), while the coupling in AB and BA regions is retained. This variation, which they term the Chirally Symmetric Continuum Model (CS-CM), highlights that the absence of AA coupling results in chiral symmetry and reveals the core properties leading to magic angles.
2. Sequence of Magic Angles: Within this CS-CM, the authors identify a sequence of magic angles where the emergence of flat bands is mathematically exact, and the bandgaps to excited states are maximized. These magic angles recur with a quasiperiodic separation in the parameter $\alpha = w_{1}/(v_{0}k_{\theta})$, approximated as 1.5. The sequence includes the first magic angle at $\theta \approx 1.09^{\circ}$, aligning closely with known experimental results.
3. Flattening of Bands and Vanishing Fermi Velocity: A vital contribution of this paper is the elucidation that the flattening of entire bands at these magic angles is intrinsically tied to the zero Fermi velocity, explained by analytic solutions of wave functions in terms of doubly-periodic theta functions. The work describes how the vanishing Fermi velocity correlates with specific vanishing points of the wavefunctions in the moiré unit cell.
4. Robustness to Perturbations: The paper shows through a perturbative analysis in $\alpha$ that the first magic angle is robust to small perturbations and a slight reinstatement of inter-layer AA coupling ($w_0$). This finding emphasizes the stability of the magic angle under modifications approaching more realistic systems and thus relates the model to practical TBG systems observed in experiments.

Implications

• Fundamental Insight into Band Flattening: This paper provides a mathematical basis for understanding the precise conditions under which flat bands emerge in TBG, offering significant implications for studies of correlated electronic phenomena, such as superconductivity, observed near these angles.
• Experimental Relevance: The results suggest that the first magic angle is attainable with negligible variance from experimentally observed angles, enhancing the practical relevance of the chirally symmetric model. This could guide future efforts in experimental design and manipulation of TBG systems to explore new quantum states of matter.
• Future Directions: The implications of this work extend to refining the theoretical models used for TBG, possibly influencing future applied research in electronic materials and devices. Moreover, understanding the interplay between these flat bands and quantum superconducting states could foster advancements in quantum computing technologies.

The paper systematically lays out calculations and theoretical findings which are likely to foster deeper exploration into the field of two-dimensional materials and could ultimately significantly influence the development of materials science and condensed matter physics.