Instabilities of coupled Cu$_2$O$_5$ ladders*

The spin-ladder compound Sr$_{14-x}$Ca$_x$Cu$_{24}$O$_{41}$ has a complex phase diagram including charge-density-wave order as well as unconventional superconductivity under high pressure. Due to its quasi-one-dimensional nature\footnote{S. Lee, J. B. Marston, J. O. Fjaerestad, Phys. Rev. B {\bf 72}, 075126.} fundamental questions about the high-T$_c$ cuprates might be more easily addressed in this context. However, due to the spatial proximity of neighboring ladders inter-ladder Coulomb repulsion as well as hopping between ladders might still be important. Using the functional renormalization group\footnote{M. Salmhofer and C. Honerkamp, Prog. Theor. Physics {\bf{105}}, 1 (2001).} and an analysis of generalized susceptibilities \footnote{D. Zanchi and H. J. Schulz, Phys. Rev. B {\bf 61}, 13609 (2000); C. J. Halboth and W. Metzner, Phys. Rev. Lett. {\bf 85}, 5162 (2000).}, we study a model of coupled Cu$_2$O$_5$ ladders \footnote{K. Wohlfeld, A. M. Oles, and G. A. Sawatzky, Phys. Rev. B {\bf 75}, 180501(R) (2007).}. We investigate instabilities towards charge, spin, and pairing order as a function of hole doping, inter-ladder hopping, and interaction strength starting from experimentally relevant hopping parameters\footnote{T. F. A. M\"u{}ller, {\it et al.}, Phys. Rev. B {\bf 57}, R12655 (1998).}.