  Motivated by recent experiments on cuprates with low-dimensional magnetic
interactions, a new class of two-dimensional Ising models with short-range
interactions and mobile defects is introduced and studied. The non-magnetic
defects form lines, which, as temperature increases, first meander and then
become unstable. Using Monte Carlo simulations and analytical low- and
high-temperature considerations, the instability of the defect stripes is
monitored for various microscopic and thermodynamic quantities in detail for a
minimal model, assuming some of the couplings to be indefinitely strong. The
robustness of the findings against weakening the interactions is discussed as
well.
