Anderson localization in Hubbard ladders

The effect of a weak random potential on two-leg Hubbard ladders is investigated. The random potential is shown to induce Anderson localization except for attractive enough interactions, supressing completely d-wave superconductivity. These localization effects remain very strong even for many ladders coupled by Josephson coupling. Both dc and ac conductivities and localization lengths are obtained. Consequences for the superconducting ladder compound Sr${}_x$Ca${}_{14-x}$Cu${}_{24}$O${}_{41+\delta }$ are discussed.