Ideal quantum geometry of the surface states of rhombohedral graphite and its effects on the surface superconductivity

The interplay of quantum geometry and interactions determines the correlated state properties of flat bands. Here, we investigate the ideal quantum geometry (IQG), i.e., the property that the trace of quantum metric equals the Berry curvature, in the surface flat bands of rhombohedrally stacked graphene (RG) multilayers. We show that RG represents a class of semimetals with IQG, among which only RG has a nonvanishing IQG at the center of the surface bands. In the presence of long-range hoppings, a displacement field polarizes the electron density to one of the two surfaces, stabilizing the IQG and boosting transitions to correlated phases like superconductivity. Analyzing the superfluid stiffness of the superconducting state in a many-layer RG, we reveal a heavy-fermion picture where the "heavy electrons" carry a nonzero supercurrent. This unusual behavior arises from the IQG of the surface states, which suggests a large fraction of supercurrent running on the surface of RG.