  We show that a doped spin-1/2 ladder with antiferromagnetic intra-chain and
ferromagnetic inter-chain coupling is a symmetry protected topologically
non-trivial Luttinger liquid. Turning on a large easy-plane spin anisotropy
drives the system to a topologically-trivial Luttinger liquid. Both phases have
full spin gaps and exhibit power-law superconducting pair correlation. The
Cooper pair symmetry is singlet $d_{xy}$ in the non-trivial phase and triplet
$S_z=0$ in the trivial phase. The topologically non-trivial Luttinger liquid
exhibits gapless spin excitations in the presence of a boundary, and it has no
non-interacting or mean-field theory analog even when the fluctuating phase in
the charge sector is pinned. As a function of the strength of spin anisotropy
there is a topological phase transition upon which the spin gap closes. We
speculate these Luttinger liquids are relevant to the superconductivity in
metalized integer spin ladders or chains.
