Out-of-equilibrium dynamics driven by localized time-dependent perturbations at quantum phase transitions

We investigate the quantum dynamics of many-body systems subject to local (i.e., restricted to a limited space region) time-dependent perturbations. If the system crosses a quantum phase transition, an off-equilibrium behavior is observed, even for a very slow driving. We show that, close to the transition, time-dependent quantities obey scaling laws. In first-order transitions, the scaling behavior is universal, and some scaling functions can be computed exactly. For continuous transitions, the scaling laws are controlled by the standard critical exponents and by the renormalization-group dimension of the perturbation at the transition. Our protocol can be implemented in existing relatively small quantum simulators, paving the way for a quantitative probe of the universal off-equilibrium scaling behavior, without the need to manipulate systems close to the thermodynamic limit.

Figure 1

The magnetization scaling function $F_O(\upsilon ,\tau )$ reported in Eq. (29) for several values of $\upsilon$. We also plot the corresponding time evolution of the ratio $M/m_0$ for the Ising ring at $g=1/2$ and $L=5$ under the protocol (19). Differences are hardly visible. Note the oscillating behavior for $\tau >0$ around the asymptotic large-$\tau$ value [38] (short horizontal dashed lines), $F_O(\upsilon ,\tau \to +\infty )=1-2e^{-\pi \upsilon /4}$; the amplitude of the oscillations slowly decreases with increasing $\tau$.

Figure 2

The scaling function $f_O(\upsilon ,\kappa )$ associated with the magnetization, cf. Eq. (30), for some values of $\upsilon$. In agreement with the OS arguments, it approaches the static limit $f_E\left(\kappa \right)=\kappa /\sqrt{1+{\kappa }^2}$ (red curve) when $\upsilon \to \infty$ keeping $\kappa$ fixed.

Figure 3

Time-dependent magnetization at the CQT, in the presence of the single-site perturbation. We plot $L^{1/8}M$ vs $\tau =t/t_s^{2/3}$ for $\upsilon \equiv t_s/L^{3/2}=1$ and several values of $L$, as obtained by numerical calculations [41]. The curves clearly approach an asymptotic function with increasing $L$, confirming the existence of OFSS. In the inset, we show $L^{1/8}M$ for $t=\tau =0$ as a function of $1/L$.

Figure 4

Plot of $\chi /\left(m_0^{2}L\right)$ vs $\tau =t/t_s^{2/3}$ for $\upsilon \equiv t_s/L^3=1$ at the magnet-kink CQT for several values of $L$. We consider the Ising Hamiltonian at $h=0$ and $g=0.5$ with the bond perturbation (43). The curves are obtained by numerical calculations [41]. They clearly approach an asymptotic function with increasing $L$, confirming OFSS. The inset shows the same quantity at $\tau =0$ vs $1/L$.