Chiral SO(4) spin-valley density wave and degenerate topological superconductivity in magic-angle twisted bilayer graphene

Abstract

Starting from a realistic extended Hubbard model for a ${p}_{x,y}$-orbital tight-binding model on the Honeycomb lattice, we perform a thorough investigation of the possible electron instabilities in magic-angle twisted bilayer graphene near the van Hove (VH) dopings. Here we focus on the interplay between the two symmetries of the system. One is the approximate $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ symmetry which leads to the degeneracy between the intervalley spin density wave (SDW) and valley density wave (VDW) as well as that between the intervalley singlet and triplet superconductivities (SCs). The other is the ${D}_{3}$ symmetry which leads to the degeneracy among the three symmetry-related wave vectors of the density-wave (DW) orders, originating from the Fermi-surface nesting. The interplay between these two degeneracies leads to intriguing quantum states relevant to recent experiments, as revealed by our systematic random-phase-approximation based calculations followed by a succeeding mean-field energy minimization for the groundstate energy. At the $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ symmetric point, the degenerate intervalley SDW and VDW are mixed into a new state of matter dubbed as the chiral SO(4) spin-valley DW. This state simultaneously hosts three four-component vectorial spin-valley DW orders with each adopting one wave vector, and the polarization directions of the three DW orders are mutually perpendicular to one another. In the presence of a tiny intervalley exchange interaction with coefficient ${J}_{H}\ensuremath{\rightarrow}{0}^{\ensuremath{-}}$ which breaks the $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ symmetry, a pure chiral SDW state is obtained. In the case of ${J}_{H}\ensuremath{\rightarrow}{0}^{+}$, although a nematic VDW order is favored, the two SDW orders with equal amplitudes are accompanied simultaneously. This nematic $\mathrm{VDW}+\mathrm{SDW}$ state possesses a stripy distribution of the charge density, consistent with the recent STM observations. On the aspect of SC, while the triplet $p+ip$ and singlet $d+id$ topological SCs are degenerate at ${J}_{H}=0$ near the VH dopings, the former (latter) is favored for ${J}_{H}\ensuremath{\rightarrow}{0}^{\ensuremath{-}}$ (${J}_{H}\ensuremath{\rightarrow}{0}^{+}$). In addition, the two asymmetric doping-dependent behaviors of the obtained pairing phase diagram are well consistent with experiments.