Scaling in driven dynamics starting in the vicinity of a quantum critical point

Driven dynamics across a quantum critical point is usually described by the Kibble-Zurek scaling. Although the original Kibble-Zurek scaling requires an adiabatic initial state, it has been shown that scaling behaviors exist even when the driven dynamics is triggered from a thermal equilibrium state exactly at the critical point, in spite of the breakdown of the initial adiabaticity. In this paper, we show that the existence of the scaling behavior can be generalized to the case of the initial state being a thermal equilibrium state near the critical point. We propose a scaling theory in which the initial parameters are included as additional scaling variables due to the breakdown of the initial adiabaticity. In particular, we demonstrate that for the driven critical dynamics in a closed system, the nontrivial thermal effects are closely related to the initial distance to the critical point. We numerically confirm the scaling theory by simulating the real-time dynamics of the one-dimensional quantum Ising model at both zero and finite temperatures.

Figure 1

Comparison of different initial conditions. Point A (gray), which is far away from the critical point $h_{\mathrm{xc}}$, signals the initial state for the original quantum KZM. The dash-dotted line (blue), which is exactly at the critical point and parallel with the vertical axis, is the starting state in the driven dynamics studied in Refs. [34, 35, 36]. In our work, the starting positions can be distributed over the whole impulse region and its finite-temperature extension (purple).

Figure 2

The evolution of $M$ under increasing $g$ with fixed $g_0{R}_g^{-1/\nu r_g}=-0.01054$ for three $R_g$ indicated. The curves before and after rescaling are shown in (a) and (b), respectively.

Figure 3

The evolution of $S$ under increasing $g$ with fixed $g_0{R}_g^{-1/\nu r_g}=-0.01054$ for three $R_g$ indicated as shown in (a). The curves for $S+(c/6r_g)log{R}_g$ versus the rescaled $g$ are shown in (b).

Figure 4

The evolution of $M$ under increasing $g$ with fixed $g_0{R}_g^{-1/\nu r_g}=0.0141,h_0{R}_g^{-\beta \delta /\nu r_g}=-0.0718$, and $T^{-1}{R}_g^{z/r_g}=0.7071$ for four $T$ indicated. The curves before and after rescaling are shown in (a) and (b), respectively.

Figure 5

(a) $M$ versus $g$ for different $g_0$ with $h_0=0.0005,R_g=0.005$, and $1/T=10$ (except the black solid curve with $1/T=\infty$ as marked). (b) $M{R}_g^{-\beta /\nu r_g}$ versus $g{R}_g^{-1/\nu r_g}$ with fixed $g_0{R}_g^{-1/\nu r_g}=-2.121$ (when $g_0=-0.15,R_g=0.005$), $h_0{R}_g^{-\beta \delta /\nu r_g}=0.07181$, and $T^{-1}{R}_g^{z/\nu r_r}=0.3535$. (c) $M{R}_g^{-\beta /\nu r_g}$ versus $g{R}_g^{-1/\nu r_g}$ with fixed $g_0=-0.15,h_0{R}_g^{-\beta \delta /\nu r_g}=0.07181$, and $T^{-1}{R}_g^{z/\nu r_r}=0.3535$.