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Charge Density Wave in Sr$_{14-x}$Ca$_x$Cu$_{24}$O$_{41}$

Recent experimental results [A. Rusydi et al., Phys. Rev. Lett. 97, 016403 (2006)] revealed the stability of a peculiar charge density wave (CDW) with odd period (three or five) in the coupled Cu$_2$O$_5$ spin ladders in Sr$_{14-x}$Ca$_x$Cu$_{24}$O$_{41}$, while the CDW with even period is unstable. Here we resolve this contradiction by solving the effective t-J model for the two coupled chains supplemented by the intersite interaction term V. Crucial to this finding is the novel interaction term V (responsible for the interladder repulsion between nearest neighbor holes), which is derived from the on-site Hubbard-like repulsion between electrons on the same oxygen sites but within two distinct orbitals which belong to two neighboring Cu$_2$O$_5$ ladders in the charge transfer model.