Time evolution of correlations in strongly interacting fermions after a quantum quench

Using the adaptive time-dependent density-matrix renormalization group, we study the time evolution of density correlations of interacting spinless fermions on a one-dimensional lattice after a sudden change in the interaction strength. Over a broad range of model parameters, the correlation function exhibits a characteristic light-cone-like time evolution representative of a ballistic transport of information. Such behavior is observed both when quenching an insulator into the metallic region and also when quenching within the insulating region. However, when a metallic state beyond the quantum critical point is quenched deep into the insulating regime, no indication for ballistic transport is observed. Instead, stable domain walls in the density correlations emerge during the time evolution, consistent with the predictions of the Kibble-Zurek mechanism.

Figure 1

(Color online) Time evolution of the equal-time density correlation function $C_{i,j}\left(t\right)$ of spinless fermions after a quench from the CDW ground state of $H\left(V_0\right)$ with $V_0=10$, evolved by the Hamiltonian $H\left(V\right)$, with (a) $V=0$, (b) $V=2$, (c) $V=5$, (d) and $V=20$.

Figure 2

(Color online) Equal-time density correlation function $C_{i,j}\left(t\right)$ of spinless fermions after a quench from the CDW ground state of $H\left(V_0\right)$ with $V_0=10$, evolved by the Hamiltonian $H\left(V\right)$, with $V=0$ and $V=2$, at fixed separations (a) $\left|i-j\right|=2$, (b) 6, (c) 10, and (d) 14 as functions of time $t$. The black vertical lines indicate the position of the horizon for $V=0$, which moves with velocity $u=4$, and the dashed vertical lines a horizon moving with velocity $u=2v\left(V=2\right)=2\pi$.

Figure 3

(Color online) Equal-time density correlation function $C_{i,j}\left(t\right)$ of spinless fermions after a quench from the CDW ground state of $H\left(V_0\right)$ with $V_0=10$, evolved by the Hamiltonian $H\left(V\right)$, with $V=5$ and $V=20$, at fixed separations $\left|i-j\right|=2$ (a), 6 (b), 10 (c), and 14 (d) as functions of time $t$ after the quench.

Figure 4

(Color online) Equal-time density correlation function $C_{i,j}\left(t\right)$ after a quench from the CDW ground state of $H\left(V_0\right)$ with $V_0=10$, evolved by the Hamiltonian $H\left(V\right)$, with $V=5$ and $V=20$, at fixed times (a) $t=0$, (b) $t=1.4$, (c) $t=2.8$, and (d) $t=4.2$ (d) as functions of the separation $\left|i-j\right|$.

Figure 5

(Color online) Equal-time density correlation function as a function of distance $\left|i-j\right|$ at different times for the quench from $V_0=10$ to $V=5$. The black lines are fits to exponential decays.

Figure 6

(Color online) Velocity of the horizon in the time evolution of the equal-time density correlation function $C_{i,j}\left(t\right)$ of spinless fermions after a quench from the CDW ground state of $H\left(V_0\right)$ with $V_0=10$, evolved by the Hamiltonian $H\left(V\right)$ as a function of $V$. The estimated error is of the order of the point size. The dashed vertical line indicates the quantum phase transition form the Luttinger liquid to the CDW regime.

Figure 7

(Color online) Time evolution of the equal-time density correlation function $C_{i,j}\left(t\right)$ after a quench from the Luttinger-liquid ground state of $H\left(V_0\right)$ with $V_0=0.5$, evolved by the Hamiltonian $H\left(V\right)$, with (a) $V=5$ and (b) $V=40$. The dashed horizontal lines indicate the time steps shown in Figs. 8b, 8c, 8d; the white circles highlight the phase slips visible there.

Figure 8

(Color online) Equal-time density correlation function $C_{i,j}\left(t\right)$ of spinless fermions after a quench from the Luttinger-liquid ground state of $H\left(V_0\right)$ with $V_0=0.5$, evolved by the Hamiltonian $H\left(V\right)$, with $V=5$ and $V=40$, at fixed times (a) $t=0$, (b) $t=1.4$, (c) $t=2.8$, and (d) $t=4.2$ as functions of the separation $\left|i-j\right|$. The blue (solid) arrows highlight the position of the phase slips obtained for $V=40$, the red (dashed) ones point to the phase slips for the case $V=5$.

Figure 9

(Color online) Local density $⟨n_i⟩\left(t\right)$ for a system of length $L=50$ with $N=25$ particles at time $t=5$ after a quench from the Luttinger-liquid ground state of $H\left(V_0\right)$ with $V_0=0.5$, evolved by the Hamiltonian $H\left(V\right)$, with (a) $V=4$ and (b) $V=40$. Arrows indicate the location of domain walls (kinks) in the local CDW pattern.

Figure 10

(Color online) Time evolution of the von Neumann entropy $S_l\left(t\right)$ for subsystems of different sizes $l=1,2,\dots ,25$, after quenches with (a) $V_0=10\to V=0$, (b) $V_0=10\to V=2$, (c) $V_0=10\to V=5$, and (d) $V_0=0.5\to V=40$. The lowest line corresponds to $l=1$, the next to $l=2$ and so on; for clarity, the curves for $l\le 5$ are labeled.