Spin and charge necklaces at commensurate filling

The charge and spin properties of spin chains decorated with dimers and closed trimers (equilateral triangles) with commensurate partial filling (1/4 and 1/3, respectively) are considered. It is shown that due to the charge separation both systems prefer the ground state with even occupation per elementary cell, where the spin spectrum possesses the Haldane gap for negative spin exchange and magnon-like for positive coupling. The charge spectrum is always gapped.