Glassy quantum dynamics of disordered Ising spins

We study the out-of-equilibrium dynamics of the quantum Ising model with power-law interactions and positional disorder. For arbitrary dimension $d$ and interaction range $\alpha \ge d$ we analytically find a stretched-exponential decay of the global magnetization and ensemble-averaged single-spin purity with a stretch power $\beta =d/\alpha$ in the thermodynamic limit. Numerically, we confirm that glassy behavior persists for finite system sizes and sufficiently strong disorder. We identify dephasing between disordered coherent pairs as the main mechanism leading to a relaxation of global magnetization, whereas genuine many-body interactions lead to a loss of single-spin purity which signifies the buildup of entanglement. The emergence of glassy dynamics in the quantum Ising model extends prior findings in classical and open quantum systems, where the stretched-exponential law is explained by a scale-invariant distribution of timescales, to both integrable and nonintegrable quantum systems.

Figure 1

(a) Magnetization decay for a uniform random spin distribution in $d=3$ with $\alpha =6$ for 1300 spins. The single-spin magnetizations $\langle {\widehat{s}}_x^i\left(\tau \right)\rangle$ for 50 different spins are shown, featuring the oscillatory behavior predicted by Eq. (2). The line color encodes the nearest-neighbor (NN) interaction strength and is thus a measure for the onset and fastest frequency of each oscillation. Its median $J_{\mathrm{NN}}$ is furthermore used as the unit for the relaxation time. Additionally, the ensemble-averaged decay is plotted (red dashed line), showing monotonous subexponential relaxation, which is well captured by a stretched-exponential function predicted analytically in the large $N$ limit (black line). Remaining deviations from the analytical solution can be attributed to the finite system size used for the simulation. (b) Histograms showing the frequency of occurrence of single-spin magnetization values at different times. The fluctuations relax on a slower timescale than the mean value, which is directly connected to the decay of the purity. (c) Single-spin purities and ensemble average, analogous to (a).

Figure 2

(a) Fit parameter $\beta$ as a function of the disorder parameter $x=N{r}_b^d/r_0^d$ for the magnetization (blue dots) and purity (green dots) for $N=100$ and $N_s=200$ for the case of $\alpha =6$ in $d=3$. At strong disorder ($x\lesssim 0.01$) $\beta$ becomes independent of $x$. Dashed lines show the averages within this regime. Error bars describe the parameter uncertainty of $\beta$ obtained from the fit. (b) Deviation of the fitted $\beta$ from the analytical solution $d/\alpha =0.5$ as a function of $N$. We average $\beta$ over five different blockade radii within the strongly disordered regime. The error bars correspond to the standard error of the mean of the latter, which is the dominant uncertainty [cf. (a)]. Dashed lines are power-law fits. (c) Fitted power-law exponent $p$ for all simulated cases $d=1,2,3$ and $\alpha =d,\cdots ,10$. The red circle highlights the points corresponding to the data shown in (b). See Supplemental Material for the choice of parameters $N$ and $N_s$.