  Low-dimensional electron systems often show a delicate interplay between
electron-phonon and electron-electron interactions, giving rise to interesting
quantum phases such as the charge density wave (CDW) and magnetism. Using the
density-functional theory (DFT) calculations with the semilocal and hybrid
exchange-correlation functionals as well as the exact-exchange plus correlation
in the random-phase approximation (EX + cRPA), we systematically investigate
the ground state of the metallic atom wires containing dangling-bond (DB)
electrons, fabricated by partially hydrogenating the GaN(10${\overline{1}}$0)
and ZnO(10${\overline{1}}$0) surfaces. We find that the CDW or
antiferromagnetic (AFM) order has an electronic energy gain due to a band-gap
opening, thereby being more stabilized compared to the metallic state. Our
semilocal DFT calculation predicts that both DB wires in
GaN(10${\overline{1}}$0) and ZnO(10${\overline{1}}$0) have the same CDW ground
state, whereas the hybrid DFT and EX+cRPA calculations predict the AFM ground
state for the former DB wire and the CDW ground state for the latter one. It is
revealed that more localized Ga DB electrons in GaN(10${\overline{1}}$0) prefer
the AFM order, while less localized Zn DB electrons in ZnO(10${\overline{1}}$0)
the CDW formation. Our findings demonstrate that the drastically different
ground states are competing in the DB wires created on the two representative
compound semiconductor surfaces.
