Spin Injection into a Luttinger Liquid

We study the effect of spin injection into a Luttinger liquid. We consider the spin-injection-detection setup of Johnson and Silsbee; here spins injected into the Luttinger liquid induce, across an interface with a ferromagnetic metal, a spin-dependent current ($I_s$) in a closed circuit or a spin-dependent boundary voltage ($V_s$) in an open circuit. We find that the spin-charge separation nature of the Luttinger liquid affects $I_s$ and $V_s$ in a very different fashion. In particular, in the Ohmic regime, $V_s$ depends on the spin transport properties of the Luttinger liquid in essentially the same way as it would in the case of a Fermi liquid. The implications of our results for prospective spin-injection-detection experiments in the high $T_c$ cuprates are discussed.