We design from first principles a type of two-dimensional metal-organic framework (MOF) using phenalenylbased ligands to exhibit a half-filled flat band of the kagome lattice, which is one of a family of lattices that show Lieb-Mielke-Tasaki's flat-band ferromagnetism. Among various MOFs, we find that trans-Au-THTAP (THTAP=trihydroxytriaminophenalenyl) has such an ideal band structure, where the Fermi energy is adjusted right at the flat band due to unpaired electrons of radical phenalenyl. The spin-orbit coupling opens a band gap giving a nonzero Chern number to the nearly flat band, as confirmed by the presence of the edge states in first-principles calculations and by fitting to the tight-binding model. This is a novel and realistic example of a system in which a nearly flat band is both ferromagnetic and topologically nontrivial.

AC transport properties of metallic carbon nanotubes with a single atomic vacancy are investigated using the Keldysh nonequilibrium Green's function method. We found that the susceptance does depend on the position of the vacancy, which makes a strong contrast to the positionindependence of the DC conductance. In addition, the susceptance indicates an inductive response in spite of the small DC conductance, when the distance between the center of the nanotube and the vacancy is larger than a certain threshold. The position dependence of the susceptance can be understood as a consequence of the breaking of parity symmetry. #

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The sub-THz AC response of metallic single-walled carbon nanotubes (M-SWNTs) with a single atomic vacancy is investigated theoretically focusing on its diameter dependence. We find that the AC response behaves more capacitively in large-diameter M-SWNTs with a vacancy at the center of the SWNT in contrast to the diameter-independent AC response of a pristine M-SWNT showing the inductive response. This can be understood from the fact that the large-diameter M-SWNTs with a vacancy have more scattering states for electrons around the vacancy than the small-diameter ones. In addition, the threshold of vacancy position from the center of the SWNT, beyond which the inductive response appears regardless of the Fermi level position, is higher for large-diameter M-SWNTs than for small-diameter ones. Moreover, we find that the AC response depends strongly on tube diameter, but not on the type of tube, i.e., armchair or zigzag. #