Magnetism of weakly hole-doped, anisotropic CuO$_2$ spin chains in La$_5$Ca$_9$Cu$_{24}$O$_{41}$

Experiments suggest that for fields $B > 4\textrm{T}$ (applied along the easy axis) the spin chain system in La$_x$Ca$_{14-x}$Cu$_{24}$O$_{41}$, $x\ge5$, is characterized by short-range spin order and quasistatic (quenched) charge disorder. Using realistic estimates for the interaction parameters, we present extensive Monte-Carlo simulations of a (classical) two-dimensional Heisenberg model with randomly distributed, static holes. We find, in particular, that the spin-flop transition of the pure model (without holes) is destroyed and transformed into a smooth anomaly. A qualitative agreement with the experimentally observed magnetic susceptibility curves in La$_5$Ca$_9$Cu$_{24}$O$_{41}$ is obtained.