Quantum Quenches in Isolated Quantum Glasses out of Equilibrium

In this work, we address the question of how a closed quantum system thermalizes in the presence of a random external potential. By investigating the quench dynamics of the isolated quantum spherical $p$-spin model, a paradigmatic model of a mean-field glass, we aim to shed new light on this complex problem. Employing a closed-time Schwinger-Keldysh path integral formalism, we first initialize the system in a random, infinite-temperature configuration and allow it to equilibrate in contact with a thermal bath before switching off the bath and performing a quench. We find evidence that increasing the strength of either the interactions or the quantum fluctuations can act to lower the effective temperature of the isolated system and stabilize glassy behavior.

Figure 1

Dynamical phase diagrams, as a function of initial temperature $T_0$ and strength of the interaction quench ${\mathcal{J}}_F/{\mathcal{J}}_0$, in two different scenarios. (a) (Left panel) The strength of quantum fluctuations is kept constant throughout the evolution, i.e., ${\hslash }_0={\hslash }_F$. In the classical case (top line) below a certain temperature the dynamics of the system displays aging. Finite quantum fluctuations suppress the aging regime (bottom line), as expected thermodynamically. (b) (Right panel) When the strength of quantum fluctuations is suddenly increased, ${\hslash }_F>{\hslash }_0$, the aging regime is enhanced (top curve) with respect to the classical phase diagram (dashed line). Vice versa, decreasing quantum fluctuations makes the aging regime shrink (bottom curve). Details of how the boundaries were obtained are given in the main text.

Figure 2

Correlation functions after the second quench for a variety of different parameters, with $N=15000$ steps, $t_{\max }=100$ and $t_q=t_{\max }/2$. In each case, ${\mathcal{J}}_0=1.0$ and the wait times are $t_w=16.67$ (dotted), $t_w=30$ (dashed), and $t_w=40$ (solid). (a) Quench of $\mathcal{J}$ in the classical (main panel) and quantum (inset) model, at $T_0=0.8$. Quenches with ${\mathcal{J}}_F<{\mathcal{J}}_0$ pump energy into the system, while quenches with ${\mathcal{J}}_F>{\mathcal{J}}_0$ extract energy and can lead to aging behavior. (b) Quench of quantum fluctuations, for $T_0=0.9$ and ${\mathcal{J}}_F=1$. A plateau emerges as ${\hslash }_F$ is increased starting from the classical limit. Vice versa, decreasing quantum fluctuations make the system thermalize rapidly (inset). (c) Role of initial temperature, for ${\mathcal{J}}_F={\mathcal{J}}_0=1$ and a sudden increase (main panel) or decrease (inset) of quantum fluctuations. In the first case lowering the temperature leads to a dynamical glass transition, consistent with the shifted phase boundary of Fig. 1.

Figure 3

Effective temperature $T_{\mathrm{eff}}$ after the second quench for a system initially equilibrated at $T_0=0.8$, ${\mathcal{J}}_F={\mathcal{J}}_0=1.0$, and ${\hslash }_0=0$, obtained from the dynamical equation and the fluctuation-dissipation relation (blue points). As the strength of quantum fluctuations is increased, $T_{\mathrm{eff}}$ decreases until it reaches approximately the dynamical temperature $T_d$ (red points, obtained from dynamical simulations in the presence of a thermal bath), here at a value ${\hslash }_F\approx 0.7$. Beyond this, $T_{\mathrm{eff}}$ displays the same nonmonotonic behavior seen in Ref. [45], indicating a violation of FDT and suggesting that the system has entered the glass phase. For comparison, the thermodynamic estimate for $T_{\mathrm{eff}}$ (see main text) is shown as a light blue line and matches almost perfectly the dynamical one. Other parameters $N=15000$ and $t_{\max }=100$, with $t_q=t_{\max }/2$.