  We investigate the low-energy properties of the orthogonal-dimer spin chain
characterized by a frustrated dimer-plaquette structure. When the competing
antiferromagnetic couplings are varied, the first-order quantum phase
transition occurs between the dimer and the plaquette phases, which is
accompanied by nontrivial features due to frustration: besides the
discontinuity in the lowest excitation gap at the transition point, a sharp
level-crossing occurs for the spectrum in the plaquette phase. We further
reveal that the plateau in the magnetization curve at 1/4 of the full moment
dramatically changes its character in the vicinity of the critical point. It is
argued that the first-order phase transition in this system captures some
essential properties found in the two-dimensional orthogonal-dimer model
proposed for $\rm SrCu_2(BO_3)_2$.
