Semiclassical approach to survival probability at quantum phase transitions

We study the decay of survival probability at quantum phase transitions with infinitely degenerate ground levels at critical points. For relatively long times, the semiclassical theory predicts power-law decay of the survival probability in systems with $d=1$ and exponential decay in systems with sufficiently large $d$, where $d$ is the degrees of freedom of the classical counterpart of the system. The predictions are checked numerically in four models.

Figure 1

A schematic plot, where $\tau$ is the time scale of interest and ${\tau }_r$ is the relaxation time, $\lambda$ is a controlling parameter with critical value ${\lambda }_c$ of a QPT. Below the solid curves, $\tau <{\tau }_r$, the system is usually not in an equilibrium state and its unitary dynamics should be considered.

Figure 2

(Color online) Decay of the SP (dashed curves) in the normal phase of Dicke model. Parameters: $\omega ={\omega }_0=1$, $ϵ={10}^{-5}$, and $\delta =-{10}^{-m}$ with $m=6,7,8,9,10,11$ from top to bottom. The solid curve is a fitting curve of the form in Eq. (5), having an initial Gaussian decay $e^{-\Gamma t^2}$ followed by a $1/\xi t$ decay. The $1/t$ decay becomes clear with increasing $m$, i.e., with ${\lambda }^{\prime }$ approaching ${\lambda }_c$. Upper right inset: $\left(\text{ln}M\right)/ϵt^2$ versus $\eta$ for different pairs of $\left(ϵ,t\right)$ with short $t$, in agreement with the prediction $\Gamma \sim \left|\eta ϵ\right|$. Lower left inset: $\text{ln}M$ versus $\text{ln}\left({ϵ}^{1/2}t\right)$ for $ϵ∊\left({10}^{-6},{10}^{-5}\right)$ and $\text{ln}t∊\left(8.6,9.5\right)$ in the $1/t$ decay region. $\delta =-{10}^{-10}$; thus, $\left|\eta \right|\simeq 1$. The results are in agreement with the prediction $\xi \sim {\left|\eta ϵ\right|}^{1/2}$.

Figure 3

(Color online) Decay of the SP (circles) in a 1-dimensional Ising chain in a transverse field, with $N=2\times {10}^8$, $ϵ=8\times {10}^{-6}$, and $\delta =-4\times {10}^{-6}$. It has the expected exponential decay (solid straight line). Lower left inset: Dependence of $\text{ln}\left(-\text{ln}M\right)$ on $\text{ln}\left|ϵ\right|$ for a fixed time $t$. The straight line has a slope 2, as predicted in Eq. (6). Upper right inset: The SP increases slowly with increasing $\left|\delta \right|$ for fixed $ϵ$ and $t$, in agreement with the prediction for $K_s$ given in the text.