Quantum quench in long-range field theories

We study the equilibration properties of observables in long-range field theories after the mass quench in $d=1$, 2, and 3 dimensions. We classify the regimes where we expect equilibration or its absence in different dimensions. In addition, we study the effect of the initial state in the equilibration properties of our system. In the case of free field theories, we show that whenever we have equilibration, the long-time limit of correlations can be described by the generalized Gibbs ensemble. We prove that all integrals of motion in our system are nonlocal.

Figure 1

Physical interpretation of the evolution of correlations after a quantum quench in the semiclassical approximation.

Figure 2

Typical plots of the two types of dispersion relations ${\omega }_1\left(k\right),{\omega }_2\left(k\right)$ and the massless case, along with the corresponding group velocity $v\left(k\right)=d\omega /dk$ as functions of $k$ for several values of the parameter $\alpha$ (2, 3/2, 1, and 1/2). The solid lines are plots of $\omega \left(k\right)$ and the dashed lines are plots of $v\left(k\right)$. The mass parameter is set equal to 1 (unless zero).

Figure 3

Dependence of the evolution of correlations on the parameter $\alpha$. The plots are density plots of the correlations [$\left|G\right(r,t\left)\right|$ as given by (12)] in space-time, for the quench from $m_0=1$ to zero mass in the type-1 dispersion relation and $1d$. The parameter $\alpha$ takes values $\frac{3}{2}$ (top) and $\frac{1}{2}$ (bottom). In the upper plot, contours are also drawn. The insets show plots of the correlation function as a function of $t$ at some fixed distances ($r=25$ and 2, respectively). The correlations increase with time for $\alpha =\frac{3}{2}$, while for $\alpha =\frac{1}{2}$ they do not. For $\alpha =\frac{1}{2}$ the system equilibrates, even though as we see in the inset, the time decay of the oscillations is very slow. Note that since the results are symmetric with respect to the vertical axis, we plotted just the positive side of the real line.

Figure 4

The effect of dimensionality on the evolution of correlations. The plots are density and contour plots of the correlations in space-time for the quench to zero mass in the type-1 dispersion relation with $\alpha =\frac{3}{2}$ and $m_0=1$ in $2d$ (top) and $3d$ (bottom). The contours become asymptotically vertical straight lines for large times, meaning that the correlations tend to stationary values. Compare with Fig. 3.

Figure 5

Density plots of correlations for the quench from massive-to-massive in the type-1 case in $1d$. The parameter $\alpha$ is $\frac{3}{2}$ (top) and $\alpha =\frac{1}{2}$ (bottom) with $m_0=2$ and $m=1$. Top: The evolution seems to exhibit a horizon, however this is not a true horizon since the decay of correlations with the distance turns out to be a power law. The dashed green line indicates the slope that corresponds to the maximum group velocity $v_{\text{max}}=\sqrt{3}/2/2^{2/3}\approx 0.5456$ for this particular value of $\alpha$, as derived from Eq. (16). Bottom: There is no sign of a horizon in this case. Even if this is not evident from the density plots due to the limited temporal range, the system equilibrates in both cases.

Figure 6

Density plot of correlations for the massive-to-massless quench in the type-2 dispersion relation with $\alpha =\frac{1}{2}$, $m_0=1$ in $1d$. There is no sign of a horizon. The inset shows a plot of the correlation function as a function of time $t$ for a fixed distance $r=2$. Compare with the bottom plot of Fig. 3, which corresponds to type 1. Notice that the decay of oscillations is much faster for type 2 than for type 1.

Figure 7

Density plots of correlations for the quench from massive-to-massive in the type-2 dispersion relation. We used the parameter values $\alpha =\frac{1}{2}$, $m_0=2$, and $m=1$ in $1d$. The plot demonstrates the existence of a causal region with a linear horizon. The dashed red line is an estimate of the horizon based on contour lines. From the slope of this line we find that the maximum group velocity is approximately $v_{\text{max}}\approx 0.14$, which is in perfect agreement with the exact value $v_{\text{max}}=3^{3/8}/7^{7/8}/2\approx 0.1375$ as derived from Eq. (17).

Figure 8

Plots of the effective mass $m^{*}$ for $m=0$ in one dimension as a function of $\alpha$ in the range from 1 to 2, for various values of $\lambda$ (0.001 to 100) in units of $m_0=1$. The results are obtained by numerical solution of the self-consistency equation (44). The solutions are real positive. Note also that the curves tend to zero in the short-range limit $\alpha =2$ with the same slope for all $\lambda$.