Quantum quench dynamics of the Coulomb Luttinger model

We study the nonequilibrium dynamics of the Luttinger model after suddenly turning on and off the bare Coulomb interaction between the fermions. We analyze several correlation functions such as the one particle density matrix and vertex correlations, its finite time dynamics, and the stationary state limit. Correlations exhibit a nonlinear light-cone effect: The spreading of the initial signal accelerates as a consequence of the quantum nature of the excitations, whose peculiar dispersion of plasmonic type in one dimension (1D) gives rise to a logarithmic divergence in the group velocity at $q=0$. In addition, we show that both the static and dynamic stationary state correlations can be reproduced with a simple generalized Gibbs ensemble despite the long-range character of the interactions which precludes the application of the Lieb-Robinson bounds. We propose a suitable experimental setup in which these effect can be observed based on ultracold ions loaded on linear traps.

Figure 1

Time evolution of ${\Phi }_1(x,t)$ from numerical integration for various values of $x$ after suddenly turning on a short-range potential (a) using the ad hoc approximation $\sinh 2{\theta }_q\approx e^{-q{R}_0}(K_f-K_f^{-1})/2$, with $R_0=d$ and $K_f=0.5$ (b) considering the full momentum dependence of a Gaussian potential Eq. (38) (${\sigma }^2=d^{-2}$) with the same Luttinger parameter $K_f$. The black dashed curve is, in both graphics, the result for $x\to \infty$. The vertical lines are, for each $x$, the times defined by Eq. (39).

Figure 2

Time evolution of ${\Phi }_1(x,t)$ for various values of $x$ after a quench from a short-range interacting initial state to a long-range interacting one with $g=1$, from numerical integration. The dashed line represents the function $\xi \left(t\right)$ defined in Eq. (35).

Figure 3

Density plot of the function ${\Phi }_1(x,t)$ for $g=1$ from numerical integration. The black dashed line is the curve $t=1.26{\tilde{t}}_x$ [Eq. (37)]. The solid red line marks the positions of the maximums. Region I is the region where the initial state correlations dominate. In region II correlations have the form of the steady state.

Figure 4

The derivative of the kinetic energy per unit length for different values of the coupling constant $g$, from numerical integration. In the inset we can see that the kinetic energy per unit length reaches a maximum and goes to a stationary state constant value. $e_{\mathrm{kin}}\left(t\right)$ is given in units of $\frac{v_F}{2\pi d^2}$.

Figure 5

Scheme of the experimental realization. Initially the ions (blue) form dipoles with its image charges (dashed circles) placed symmetrically with respect to the gate (thick solid line). The dipoles interact with each other through the dipole interaction, decaying as $x^{-3}$. After suddenly removing the gate the ions interact via the bare Coulomb repulsion.