Optical Signatures of Band Flatness and Anisotropic Quantum Geometry in Magic-Angle Twisted Bilayer Graphene

Abstract

We study the degree of band flatness and anisotropic quantum geometry in magic-angle twisted bilayer graphene by varying the twist angle and the lattice relaxation through optical conductivity. We show that the degree of band flatness and its quantum geometry can be revealed through optical absorption and its resulting optical bounds, which are based on the trace condition in quantum geometry. More specifically, the narrow and isolated peak of optical absorption in the low-energy region provides information about the bandwidth between two flat bands. When this value is smaller than the electron interaction, it serves as a critical condition for the emergence of flat band superconductivity. Furthermore, optical absorption also provides the gap value between the flat band and the dispersive band, and when this gap is larger than the electron interaction, it facilitates the realization of fractional Chern insulating phases. We show that the narrow and isolated peak of optical bound near zero energy decreases as lattice relaxation increases. Meanwhile, we demonstrate that the imaginary part of generalized optical Hall conductivity reveals the vanishing of the negative part of Berry curvature, which is enforced by the refined trace-determinant inequality. Accordingly, we show that the total amount of the negative part and component of the Berry curvature approaches zero in the single ideal flat-band case. In contrast, when considering all occupied bands, the total amount of the negative component is slightly different from zero. Finally, we demonstrate that the condition of vanishing of flat band velocities and the emergent chiral symmetry are sufficient for the saturation of the trace condition, which pertains to the isotropic case.

Figures (6)

• Figure 1: Band structure and optical conductivity as a function of the selected twist angle $\theta$. (a1-a3) Band structure of TBG with twist angle $\theta=1.2^{\circ},1.1^{\circ},1.0^{\circ}$. The color bands indicate the strength of Coulomb interactions extracted from STM experiments Xie19Torma22. (b1-b3) Optical conductivity with three regions of twist angles ($\theta_{c1}>\theta\ge1.2^{\circ}, 1.2^{\circ}>\theta>1.0^{\circ}, \theta_{c2}<\theta\le1.0^{\circ}$) in units of $\sigma_{0}=\frac{e^2}{h}$. Here we plot the optical conductivity with two twist angles in each region to illustrate the trend. The dashed black line marked in (b1-b3) represents the strength of Coulomb interaction extracted from STM data Xie19Torma22. Note that experimentally, superconductivity was found with twist angle in the region $1.2^{\circ}>\theta>1.0^{\circ}$Balents20, where the bandwidth, shown in the first zone bounded by zeros in the optical conductivity, is smaller than the strength Coulomb interaction. Here we set $\kappa = 0.8$, and $(\Delta_b, \Delta_t) = (5, 0)$ meV, which slightly opens a gap; the other parameters are the same as in the main text.
• Figure 2: Band structure, trace condition, and optical bound (inequality) as a function of $\kappa=w_{0}/w_{1}$ with $\theta=1.05^{\circ}$. (a1-a4) Band structure of TBG with $\kappa=0.8,0.6,0.4,0.0$. The color bands indicate the strength of Coulomb interactions extracted from STM experiments Xie19Torma22. (b1-b4) Trace condition of the uppermost valence flat band with the corresponding values of $\kappa$. (c1-c4) Optical bound and imaginary part of generalized optical Hall conductivity with the corresponding value of $\kappa$, in units of $\sigma_{0}=\frac{e^2}{h}$. The red (green dashed) line represents the optical bound (imaginary part of generalized optical Hall conductivity). Note that because some values of $|\Omega_{xy}^{\nu}|$ are close to zero, we consider the subtraction form of the trace condition when $\kappa=0.8$.
• Figure 3: (a) The negative part of the Berry curvature of the valence flat and occupied band as a function of $\kappa$. Depending on whether $F^{-}_{\nu}$ and/or $F^{-}_{occ}$ reach zero, we divide the range of $\kappa$ into three colored regions. (b) The ratio of total amounts of the negative and positive components of Berry curvature of valence flat and occupied band as a function of $\kappa$. (c) The ratio of $g_{xx}^{\nu}$ and $g_{yy}^{\nu}$ with $\kappa=0$, is shown over the MBZ. (d) The off-diagonal component of the quantum metric, $g_{xy}^{\nu}$, with $\kappa=0$, is shown over the MBZ. Note that the values of $g_{xx}^{\nu}/g_{yy}^{\nu}$ and $g_{xy}^{\nu}$, close to one and zero respectively, imply the isotropic FCI phase. Here we set $\theta=1.05^{\circ}$. The evolution of the quantum metric as a function of $\kappa$ is shown in the SM supp.
• Figure S1: Evolution of the valence flat band's Abelian quantum metric and its trace condition as varying lattice relaxation, with $\theta=1.05^{\circ}$. (a1-a4) Evolution of the Abelian quantum metric ratio, $g_{xx}^{\nu}/g_{yy}^{\nu}$. (b1-b4) Evolution of the off-diagonal Abelian quantum metric element, $g_{xy}^{\nu}$. (c1-c4) Evolution of the trace condition of the valence flat band. Note that because some values of $|\Omega_{xy}^{\nu}|$ are close to zero, we consider the subtraction form of the trace condition when $\kappa=0.8$.
• Figure S2: Evolution of the quantum metric of all occupied bands and its trace condition as varying lattice relaxation, with $\theta=1.05^{\circ}$. (a1-a4) Evolution of the non-Abelian quantum metric ratio, $g_{xx}/g_{yy}$. (b1-b4) Evolution of off-diagonal non-Abelian quantum metric element, $g_{xy}$. (c1-c4) Evolution of the trace condition of all occupied bands. Note that because some values of $|\Omega_{xy}^{\nu}|$ are close to zero, we consider the subtraction form of the trace condition when $\kappa=0.8$.