Interacting valley Chern insulator and its topological imprint on moiré superconductors

One salient feature of systems with moiré superlattice is that the Chern number of “minibands” originating from each valley of the original graphene Brillouin zone becomes a well-defined quantized number because the miniband from each valley can be isolated from the rest of the spectrum due to the moiré potential. Then a moiré system with a well-defined valley Chern number can become a nonchiral topological insulator with $\mathrm{U}\left(1\right)\times Z_3$ symmetry and a $\mathbb{Z}$ classification at the free fermion level. Here we demonstrate that the strongly interacting nature of the moiré system reduces the classification of the valley Chern insulator from $\mathbb{Z}$ to ${\mathbb{Z}}_3$, and it is topologically equivalent to a bosonic symmetry-protected topological state made of local boson operators. We also demonstrate that even if the system becomes a superconductor when doped away from the valley Chern insulator, the valley Chern insulator still leaves a topological imprint as the localized Majorana fermion zero mode in certain geometric configuration.

Figure 1

(a) The proposed experimental geometry for detecting the Majorana zero mode (MZM). The entire system is the TDBG in its spin-polarized superconductor phase near half-filling away from charge neutrality [7, 9, 10]. Displacement field (out of plane) is adjusted such that the valley Chern numbers of the filled minibands differ by 1 between the two domains. The vertical domain wall preserves the valley quantum number, while the horizontal domain wall breaks the valley quantum number conservation. We propose that a MZM is located at the $0d$ junction between the two domain walls. (b) The geometry of our effective lattice model. The $1d$ domain walls in (a) are modeled by the open boundary of Eq. (12) defined on a cylinder with size $20\times 20$. The horizontal domain wall in (a) is modeled by an intervalley scattering $M{\sigma }^{211}$ on half of the boundary.

Figure 2

The band structure of our effective lattice model for each valley with two open boundaries. As we can see there are two counterpropagating edge modes from the two boundaries. We have chosen $t_1=0.3, t_2=0.2$, and $m=0.05$ in our effective model, Eq. (12). The horizontal line is the energy (chemical potential) $\mu =0.5$.

Figure 3

The low-energy modes of our effective lattice model plotted against chemical potential, with $s$-wave valley-singlet pairing amplitude $\mathrm{\Delta }=0.2$ in the bulk, and intervalley scattering $M=0.8$ on half of the boundary. We observe two MZMs on the two $0d$ junctions between $M$ and $\mathrm{\Delta }$ at the boundary, for a broad range of finite chemical potential, even when the bulk has finite charge carrier density.

Figure 4

(a) Penetration of the spatial wave function of the edge states (Fig. 2) into the bulk of Eq. (12), plotted against momentum $k_1$ along the boundary. (b) Spatial wave function of MZMs at the junctions between $\mathrm{\Delta }$ and $M$ at the open boundary of our model [Fig. 1], with chemical potential $\mu =0.5, \mathrm{\Delta }=0.2$, and $M=0.8$.