Steady states and universal conductance in a quenched Luttinger model

We obtain exact analytical results for the evolution of a 1+1-dimensional Luttinger model prepared in a domain wall initial state, i.e., a state with different densities on its left and right sides. Such an initial state is modeled as the ground state of a translation invariant Luttinger Hamiltonian $H_{\lambda }$ with short range non-local interaction and different chemical potentials to the left and right of the origin. The system evolves for time $t>0$ via a Hamiltonian $H_{{\lambda }^{\mathrm{\prime }}}$ which differs from $H_{\lambda }$ by the strength of the interaction. Asymptotically in time, as $t\to \mathrm{\infty }$, after taking the thermodynamic limit, the system approaches a translation invariant steady state. This final steady state carries a current $I$ and has an effective chemical potential difference ${\mu }_{+}-{\mu }_{-}$ between right- ($+$) and left- ($-$) moving fermions obtained from the two-point correlation function. Both $I$ and ${\mu }_{+}-{\mu }_{-}$ depend on $\lambda$ and ${\lambda }^{\mathrm{\prime }}$. Only for the case $\lambda ={\lambda }^{\mathrm{\prime }}=0$ does ${\mu }_{+}-{\mu }_{-}$ equal the difference in the initial left and right chemical potentials. Nevertheless, the Landauer conductance for the final state, $G=I\mathrm{/}({\mu }_{+}-{\mu }_{-})$, has a universal value equal to the conductance quantum $e^2\mathrm{/}h$ for the spinless case.