Theoretical Analysis on Pseudo-Degenerate Zero-Energy Modes in Vacancy-Centered Hexagonal Armchair Nanographene

Deriving mathematical expressions of two zero modes for a $\pi$-band tight-binding model, we identify a class of bipartite graphs having the same number of subgraph sites, where each graph represents one of the quasi-hexagonal nanographene molecule with a center vacancy (VANG). Indeed, in a VANG molecule, C$_{60}$H$_{24}$, showing stability in a density-functional simulation at the highest occupied level, there appear two pseudo-degenerate zero modes, a vacancy-centered quasi-localized zero mode, and extending zero mode with a $\sqrt{3} \times \sqrt{3}$ structure. Since there is a finite energy gap between these two zero-energy modes and the other modes, low-lying states composed of quasi-degenerate zero modes appear as magnetic multiplets. Thus, the unique magnetic characteristics derived in our theory are expected to hold for synthesized VANG molecules in reality.