Neutral Magic-Angle Bilayer Graphene: Condon Instability and Chiral Resonances

Abstract

The full optical response of twisted bilayer graphene at the neutrality point close to the magic angle within the continuum model (CM) is discussed. First, three different channels consistent with the underlying $D_3$ symmetry are identified, yielding the total, magnetic, and chiral response. Second, the full optical response in the immediate vicinity of the magic angle ${\theta }_m$ is numerically calculated, which provides a direct mapping of the CM onto an effective two-band model. It is, further, shown that the ground state of the CM in the immediate vicinity of ${\theta }_m$ is unstable toward transverse current fluctuations, a so-called Condon instability. Third, due to the large counterflow, the acoustic plasmonic excitations with typical wave numbers have larger energies than the optical ones and their energy density may be largely enhanced at certain frequencies which are denominated as chiral resonances. Finally, symmetry relations for the optical response and their consequences for the chiral response are discussed.

Keywords: chiral response; optical absorption; plasmons; twisted bilayer graphene.

Figure 1

Dissipative response of the total (left), magnetic (center), and chiral (right) current for the symmetric continuum model $\kappa =1$ at various twist angles around the magic angle ${\theta }_{\mathrm{m}}\simeq {1.03}^{\circ }$. The real conductivities $\text{Re}{\sigma }_{\text{tot}}$, $\text{Re}{\sigma }_{\text{mag}}$, and $\text{Re}{\sigma }_{\text{chi}}$ are given in units of the universal absorption of single‐layer graphene, ${\sigma }_{\mathrm{G}}=\frac{{\mathrm{e}}^2}{4\hslash }$. The inset of the left panel shows the universal absorption of two graphene layers, $2{\sigma }_{\mathrm{G}}$, independent of the twist angle.

Figure 2

Reactive response of the total (left), magnetic (center), and chiral (right) current for the symmetric continuum model with $\kappa =1$ in Equation (3) at various twist angles around the magic angle ${\theta }_{\mathrm{m}}\simeq {1.03}^{\circ }$. The real current susceptibilities $\text{Re}{\chi }_{\text{tot}}$, $\text{Re}{\chi }_{\text{mag}}$, and $\text{Re}{\chi }_{\text{chi}}$ are given in units of $t\frac{{\mathrm{e}}^2}{{\hslash }^2}$ with $t=2.78\text{eV}$.

Figure 3

Band structure of the two flat bands around charge neutrality of the continuum model with $\kappa =1$ in Equation (3) for various twist angles around the magic angle ${\theta }_{\mathrm{m}}\approx {1.032}^{\circ }$. In the left panel, the band structure with smallest bandwidth is shown. In the center panel, one can observe the avoided crossings along the KM‐direction, whereas in the right panel the avoided crossings are along the $\Gamma \mathbf{\text{k}}$‐direction; see insets. The arrows indicate the avoided crossings for $\theta {=1.05}^{\circ }$ and $\theta {=1.02}^{\circ }$.

Figure 4

The optical response for twist angles below the magic angle ${\theta }_{\mathrm{m}}{=1.032}^{\circ }$. The insets show the logarithm of the Dirac regime (left) and the optical response functions ${\sigma }_{\text{mag}}$ (center) and ${\sigma }_{\text{chi}}$ (right) at $\omega =0$ as function of the effective parameter lnα defined in Equation (26).

Figure 5

The real part of the current susceptibility $\text{Re}{\chi }_{\nu }(\omega )$ with $\nu =\text{tot},\text{mag},\text{chi}$ of the asymmetric continuum model with $\kappa =0.8$ in Equation (2) at the neutrality point in units of $t\frac{{\mathrm{e}}^2}{{\hslash }^2}$ for temperatures $T=\mathrm{0,10}\mathrm{K}$. The optical gap is indicated by the white area. Left: twist angle $\theta {=1.3}^{\circ }$. Center: twist angle $\theta {=1.2}^{\circ }$. Right: twist angle $\theta {=1.1}^{\circ }$.

Figure 6

Left: Illustration of the detailed balance relation of a particle–hole symmetric mode. Via the antiunitary transformation $\mathcal{U}$, the transitions from ${\epsilon }_n\to {\epsilon }_{\mathrm{m}}$ at momentum $(k_x,k_y)$ are directly related to the transitions from ${\epsilon }_{\tilde{\mathrm{m}}}\to {\epsilon }_{\tilde{n}}$ at momentum $(-k_x,k_y)$. Any hole transition (${\epsilon }_n^h>{\epsilon }_{\mathrm{m}}^{\mathrm{e}}$) is automatically related to an electron transition (${\epsilon }_{\tilde{n}}^e>{\epsilon }_{\tilde{m}}^h$) because ${\epsilon }_{\tilde{n}}^e={\epsilon }_n^h$ and ${\epsilon }_{\tilde{m}}^h={\epsilon }_m^e$. Center: Chiral response $\text{Re}{\sigma }_{\text{chi}}(\omega )$ of the asymmetric continuum model with $\kappa =0.8$ in Equation (2) at the neutrality point with twist angle $\theta {=1.1}^{\circ }$ for temperatures $T=\mathrm{0,10,300}\mathrm{K}$. The inset highlights the chiral response around $\hslash \omega =95\text{meV}$. Right: Corresponding band structure and DOS on logarithmic scale. The transitions related to the van Hove singularities around $\hslash \omega =25\text{meV}$ and $\hslash \omega =95\text{meV}$ are indicated by red (electronic transition) and blue (hole‐like transition) arrows.