Steady States and Universal Conductance in a Quenched Luttinger Model

Abstract

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 $H_{\lambda'}$ which differs from $H_{\lambda}$ by the strength of the interaction. Asymptotically in time, as $t \to \infty$, 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 $\lambda'$. Only for the case $\lambda = \lambda' = 0$ does $\mu_+ - \mu_-$ equal the difference in the initial left and right chemical potentials. Nevertheless, the Landauer conductance for the final state, $G=I/(\mu_+ - \mu_-)$, has a universal value equal to the conductance quantum $e^2/h$ for the spinless case.