Time-Dependent Evolution of Two Coupled Luttinger Models of Spin-1/2 Fermions

Based on the field-theoretical bosonization we consider the spinful system consisting of two disconnected Luttinger liquids, which are coupled at time t = 0 through chiral density-density interactions. We show the well-known spin-charge separation in this ID system by means of bosonization and Bogoliubov transformation. To evaluate analytically the one-particle equal-time correlation functions, Cazalilla's methods are extended to spin-charge sectors. It is found that in long time limit the behavior of correlations does not coincide with that of equilibrium equal-time correlations of two coupled spinful Luttinger liquids. In order to compare with experimental results, the improved exponent that governs the power-law behavior of equal-time correlations of the non-equilibrium system is obtained at the end of this work.