Drumhead Surface States of Rhombohedral Graphite with Near Ideal Quantum Geometry Condition

Abstract

The ideal quantum geometry (IQG) condition of equal magnitude between quantum metric ($G(k)$) and Berry curvature ($Ω(k)$) as in $|Ω|$/Tr$G$=1 has been associated with realizing fractional Chern insulators for the flat band in few layers of rhombohedral graphene (RG). More recently the IQG condition has also been proposed for superconductivity in the flat band for thick RG layers. Using density functional theory and Wannier functions, we study the symmetry-protected topology of bulk RG, drumhead surface state (DSS) of semi-infinite RG surface, and the IQG condition for thick RG slabs. We find that bulk RG is a weak topological insulator with spin-orbit coupling (SOC), besides being effectively a chiral semimetal with chiral nodal line without SOC. We also find that the DSS flat band of semi-infinite and thick RG slabs have a sizable convex curvature with depth agreeing with the recent angular resolved phono-emission spectroscopy experiment. The calculated IQG also shows a convex shape with the K point at the center having a minimum with approximately $|Ω|$/Tr$G$$\approx$1. But the inner rim of the DSS region shows the strict IQG condition of $|Ω|$/Tr$G$=1. These results on semi-infinite and thick RG slabs from first-principles calculations provide useful information for the pristine DSS at the single particle level for future studies to consider when many-body interactions and strong correlations will be included.