Chiral Spin Density Wave and $d+id$ Superconductivity in the Magic-Angle-Twisted Bilayer Graphene

We model the newly synthesized magic-angle-twisted bilayer graphene superconductor with two $p_{x,y}$-like Wannier orbitals on the superstructure honeycomb lattice, where the hopping integrals are constructed via the Slater-Koster formulism by symmetry analysis. The characteristics exhibited in this simple model are well consistent with both the rigorous calculations and experiment observations. A van Hove singularity and Fermi-surface (FS) nesting are found in the doping levels relevant to the correlated insulator and unconventional superconductivity revealed experimentally, based on which we identify the two phases as weak-coupling FS instabilities. Then, with repulsive Hubbard interactions turned on, we performed random-phase-approximation based calculations to identify the electron instabilities. As a result, we find chiral $d+id$ topological superconductivity bordering the correlated insulating state near half-filling, identified as noncoplanar chiral spin-density wave ordered state, featuring the quantum anomalous Hall effect. The phase diagram obtained in our approach is qualitatively consistent with experiments.

Figure 1

(a) Schematic diagram for our model. The dashed rhombus labels the unit cell of the emergent honeycomb lattice with the $p_x$, $p_y$-like Wannier orbitals on each site. (b) Band structure and (c) DOS of MA-TBG. The red, black, and green horizontal dashed lines in (c) label the filling $\delta$ of $-\frac{1}{2}$, 0, and $\frac{1}{2}$, respectively. The Slater-Koster parameters $t_{\sigma /\pi }^{ij}$ are chosen as $t_{\sigma }^{(1)}=2\mathrm{meV}$, $t_{\pi }^{(1)}=t_{\sigma }^{(1)}/1.56$, $t_{\sigma }^{(2)}=t_{\sigma }^{(1)}/7$, and $t_{\pi }^{(2)}=t_{\sigma }^{(2)}/1.56$. Here, the superscript $(1)/(2)$ represents NN or NNN bondings, respectively.

Figure 2

The evolution of FS. (a)–(c) FS at fillings $-0.40$, $-0.425$ (${\delta }_V$), and $-0.45$. (d)–(f) FS at fillings 0.41, 0.425 (${\delta }_V$) and 0.44. The central hexagons in every plot outlined with black dashed lines indicate the Brillouin Zone. The FS-nesting vectors ${\mathit{Q}}_{\alpha }$ ($\alpha =1$, 2, 3) are marked in (b) and (e).

Figure 3

Distribution of $\chi (\mathit{k})$ for $\delta \to {\delta }_V=-0.425$ (a) in the Brillouin Zone, and (b) along the high-symmetric lines. (c) The filling dependence of $U_c$ (for $J_H=0.1U$), with the horizontal line representing $U=1.5\mathrm{meV}$ adopted in our calculations.

Figure 4

The doping dependence of the largest eigenvalues $\lambda$ of all pairing symmetries for (a) $J_H=0.1U$ and (b) $J_H=0$. Note that the shown eigenvalue for the $f$ wave is the larger one of the two different $f$ symmetries. The vertical bold grey lines indicate the SDW regime. (c) and (d) are the gap functions of $d_{x^2-y^2}$ and $d_{xy}$-wave symmetries at doping $\delta =-0.5$, respectively.