Moore-Read state in half-filled moiré Chern band from three-body pseudopotential

The moir\'e system provides a tunable platform for exploring exotic phases of materials. This article shows the possible realization of a non-Abelian state characterized by the Moore-Read wavefunction in a half-filled moir\'e Chern band, exemplified by twisted ${\mathrm{MoTe}}₂$. This is achieved by introducing short-range repulsive three-body interaction. Exact diagonalization is employed to examine the spectrum in finite size. The incompressibility of the system, the degeneracy of the ground states, and the number of low-energy states provide compelling evidence to identify the ground state as the Moore-Read state. We further interpolate between the three-body interaction and Coulomb interaction to show a phase transition between the composite Fermi-liquid and the Moore-Read state. Finally, we explore the conditions under which the three-body pseudopotential from band mixing effects mimic short-range repulsive three-body interaction. We provide insights toward realizing non-Abelian phases of matter in the moir\'e system.