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 $d_{\mathit{xy}}$ 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.

我们表明，具有反铁磁链内和铁磁链间耦合的掺杂自旋 1/2 阶梯是一种对称保护的拓扑非平凡 Luttinger 液体。打开一个大的易平面自旋各向异性将系统驱动到拓扑平凡的 Luttinger 液体。两相都具有完整的自旋间隙并表现出幂律超导对相关性。Cooper 对对称性是单线态$d_{\mathit{xy}}$在非平凡阶段和三重态${\mathrm{小号}}_z=0$在琐碎的阶段。拓扑上非平凡的 Luttinger 液体在存在边界的情况下表现出无间隙自旋激发，并且即使在电荷扇区中的波动相位被固定时，它也没有非相互作用或平均场理论模拟。作为自旋各向异性强度的函数，存在自旋间隙闭合的拓扑相变。我们推测这些 Luttinger 液体与金属化整数自旋梯或链中的超导性有关。