Tunable Bi-frustrated Electron Spin and Charge States in a Cu-Hexaaminobenzene Framework*

A geometrically frustrated lattice may host frustrated electron spin or charge states that spawn exotic quantum phases. We show that a newly synthesized metal-organic framework of Cu-Hexaaminobenzene [Cu₃(HAB)₂] exhibits a multi spectra of unusual quantum phases long sought after in condensed matter physics. On one hand, the Cu²⁺ ions form an ideal S-1/2 antiferromagnetic kagome lattice. On the other hand, the conjugated-electrons from the organic ligands give rise to completely dispersionless energy bands around the Fermi level, reproducing a frustrated π_x-π_y hopping model on a honeycomb lattice. We propose to characterize the coexistence of frustrated local spins and conjugated electrons through scanning tunneling microscopy simulations. Most remarkably, their close energy proximity enables one to tune the system between the two frustrated states by doping up to one hole per HAB unit. Thus, Cu₃(HAB)₂ provides a unique exciting platform to investigate the interplay of frustrated spins and electrons in one single lattice, e.g. by gating experiments, which will undoubtedly raise interesting theoretical questions leading to possible new condensed matter phases.