Light-cone velocities after a global quench in a noninteracting model

We study the light-cone velocity for global quenches in the noninteracting XY chain starting from a class of initial states that are eigenstates of the local $z$ component of the spin. We point out how translation invariance of the initial state can affect the maximal speed at which correlations spread. As a consequence the light-cone velocity can be state dependent also for noninteracting systems: a new effect of which we provide clear numerical evidence and analytic predictions. Analogous considerations, based on numerical results, are drawn for the evolution of the entanglement entropy.

Figure 1

On the right panel ${\mathrm{\Delta }}_{ln}\left(t\right)$ for the XX chain ($\gamma =h=0$) and $J=-1$ with different initial states; here we have $L=144$ and $|l-n|=60$. Vertical bars in correspondence of $\frac{|l-n|}{2v_{max}}$. On the left panel we plotted the numerically estimated inflection points $\tau$ of the signals as a function of the distances $|l-n|$. The points align on a straight line with slope given in Eq. (32).

Figure 2

On the right panel, ${\mathrm{\Delta }}_{ln}\left(t\right)$ for the XY chain and $J=1$ starting from the state $(1/2,1)$ with vertical bars in correspondence of $\frac{|l-n|}{2v_{max}}$; here $L=96$ and $|l-n|=12$. On the left panel we plotted the numerically estimated inflection points $\tau$ of the signals as a function of the distances $|l-n|$. The points align on a straight line with slope obtained from Eq. (35).

Figure 3

$|C_{ln}{|}^2, |F_{ln}{|}^2$, and ${\mathrm{\Delta }}_{ln}\left(t\right)$ for the XY chain with $J=1, \gamma =1$, and $h=1$ (critical Ising chain). The continuous vertical lines intersecting the red and blue curves are indicating the inflection points $\tau$. The dashed vertical line is in correspondence of the inflection point of ${\mathrm{\Delta }}_{ln}$ (green curve). It can be seen that $|F_{ln}|$ and $|C_{ln}|$ propagates faster than ${\mathrm{\Delta }}_{ln}$. Here $L=144$ and $|l-n|=60$.

Figure 4

The continuous blue curve is the function $-{\text{Ai}}^2(-X)$. The points represent numerical evaluations of ${\left(4t\right)}^{2/3}{\mathrm{\Delta }}_{ln}\left(t\right)$ for a quench from the Néel state ($p=2$) in the critical Ising chain ($h=\gamma =1$ and $J=1/2$). Here $|l-n|=2v_{\text{max}}t-{[-t{e}^{\prime \prime \prime }\left(k_{max}\right)]}^{1/3}X$ and $t=500$.

Figure 5

The evolution of the entanglement entropy in the XY chain with system size $L=600$ and $l=72$. The vertical dashed lines are in correspondence of $t=\frac{l}{2v_{max}}$, being $v_{max}$ the light-cone velocity for the correlation functions obtained in Sec. 4. (a) $S\left(t\right)$ at $\gamma =2$ and $h=1.5$. The blue curve corresponds to time evolution from $p=1$ (i.e., a fully polarized initial state) and the green one to $p=3$. Both are compatible with ${\tau }_s=\frac{l}{2v_{max}}$, notice that for $p=1v_{max}=v_g$. The red curve corresponds instead to $p=2$ and shows that in such a case the saturation time is larger than $\frac{l}{2v_{max}}$. (b) $S\left(t\right)$ in the free fermion case $\gamma =h=0$ and $J=-1$, here the saturation time is in agreement with the conjecture ${\tau }_s=\frac{l}{2v_{max}}$.

Figure 6

From top to bottom. First panel: Maximum group velocity $v_{max}$, obtained from Eq. (35) and $v_g$ for different values of $\gamma$ and $h$ at $p=2$. Second panel: The difference $d$ between $v_{max}$ and $v_g$ at $p=2$. Here we took $J=1$.