Charge order in the incommensurate compounds $Sr_{14-x}Ca_xCu_{24}O_{41}$

The present paper studies, using ab-initio calculations, the influence of the incommensurate structural modulations on the low energy physics of the $Sr_{14-x}Ca_xCu_{24}O_{41}$ oxides. On-site, nearest-neighbor and next-nearest-neighbor effective parameters were computed within a $t-J+V$ model based on the copper magnetic orbitals. The structural modulations appear to be the key degree of freedom, responsible for the low energy properties such as the electron localization, the formation of dimers in the $x=0$ compound or the anti-ferromagnetic order in the $x=13.6$ compound