Transport properties for a Luttinger liquid wire with Rashba spin-orbit coupling and Zeeman splitting

We study the transport properties for a Luttinger liquid ( LL) quantum wire in the presence of both Rashba spin - orbit coupling ( SOC) and a weak external in-plane magnetic field. The bosonized Hamiltonian of the system with an externally applied longitudinal electric field is established. Then the equations of motion for the bosonic phase fields are solved in Fourier space, with which both the charge and spin conductivities for the system are calculated analytically, based on linear response theory. Generally, the ac conductivity is an oscillation function of the strengths of electron - electron interaction, Rashba SOC and magnetic field, as well as the driving frequency and the measurement position in the wire. Through analysis with some examples it is demonstrated that the modification of the conductivity due to electron - electron interactions is more remarkable than that due to SOC, while the effects of SOC and Zeeman splitting on the conductivity are very similar. The spin-polarized conductivities for the system in the absence of Zeeman effect or SOC are also discussed. The ratio of the spin- polarized conductivities sigma(up arrow)/sigma(down arrow). is dependent on the electron electron interactions for a system without SOC, while it is independent of the electron - electron interactions for a system without Zeeman splitting.