Pair density wave and loop current promoted by Van Hove singularities in moiré systems

We theoretically show that in the presence of conventional or higher order van Hove singularities(VHS), bare finite momentum pairing, also known as pair density wave (PDW), susceptibility can be promoted to same most divergent BCS through a valley-contrasting flux 3$\phi$ each triangular plaquette at $\phi=\frac{\pi}{3}$ and $\phi=\frac{\pi}{6}$ moir\'e systems. This makes PDW possible leading instability for an electronic system with repulsive interactions. confirm it indeed wins over all other instabilities becomes ground state under certain conditions renormalization group calculation insertion argument. Moreover, we find topological nontrivial loop current if Fermi surface VHS is perfectly nested $\phi=\frac{\pi}{3}$. Similar Haldane model, this has quantum anomalous Hall effect. If dope introduce next-nearest neighbour hopping $t^{\prime}$, chiral $d$-wave dominant instability. Experimentally, effectively tuned by out-of-plane electric field systems based on graphene transition metal dichalcogenides.