Strong and Weak Thermalization of Infinite Nonintegrable Quantum Systems

When a nonintegrable system evolves out of equilibrium for a long time, local observables are in general expected to attain stationary expectation values, independent of the details of the initial state. But the thermalization of a closed quantum system is not yet well understood. Here we show that it presents indeed a much richer phenomenology than its classical counterpart. Using a new numerical technique, we identify two distinct regimes, strong and weak, occurring for different initial states. Strong thermalization, intrinsically quantum, happens when instantaneous local expectation values converge to the thermal ones. Weak thermalization, well known in classical systems, shows convergence to thermal values only after time averaging. Remarkably, we find a third group of states showing no thermalization, neither strong nor weak, to the time scales one can reliably simulate.

Figure 1

Strong thermalization: initial state $|Y+⟩$. The main plot shows the distance between the three-body reduced density matrix (3-RDM) at time $t$ and the corresponding thermal state ($\beta =0$). Error bars show the difference between results with bond dimension $D=120$ and $D=60$ (see the Appendix). The right-hand inset shows the distance between the time-averaged reduced density matrix and the thermal state. The solid line shows a fit to a curve whose asymptotic behavior at long times is $b/\sqrt{t}$, with $b=0.047$. The left-hand inset shows the difference between thermal expectation values and time-dependent single-body observables $⟨{\sigma }_x⟩$ (blue solid line), $⟨{\sigma }_y⟩$ (dashed green line), and $⟨{\sigma }_z⟩$ (dash-dotted black line). Convergence to the thermal state is observed in all plots.

Figure 2

Weak thermalization: initial state $|Z+⟩$. The distance between the 3-RDM and the thermal state of the same energy ($\beta =0.7275$) oscillates strongly with time. The right-hand inset shows the distance between the time-averaged reduced density matrix and the thermal ensemble. The superimposed solid curve, which behaves as $b/\sqrt{t}$ in the long-time limit ($b=0.144$ from the fit), shows that the average behavior is compatible with a slow convergence. The left-hand inset shows the oscillations of individual single-body time-dependent expectation values around thermal ones. No damping is observed for the longest simulated times ($t\sim 18$). The error bars give the distance between results for $D=240$ (shown) and $D=120$.

Figure 3

No thermalization observed: initial state $|X+⟩$. The plot shows the distance between the evolved 3-RDM and the thermal state ($\beta =-0.7180$). No thermalization is observed in the instantaneous or in the time-averaged (right-hand inset) density matrix. On the latter, we superimpose a fit to a time-dependent function, which asymptotically tends to a constant (0.03). The error bars give the distance between results for $D=240$ (plotted) and $D=120$. The left-hand inset shows the distance to the thermal values for the single-body observables, $⟨{\sigma }_x⟩$ (blue solid line), $⟨{\sigma }_y⟩$ (dashed green line), and $⟨{\sigma }_z⟩$ (dash-dotted black line). We observe that $⟨{\sigma }_x⟩$ is responsible for the lack of thermalization, while the others converge to thermal.

Figure 4

Distance between the averaged three-body reduced density matrix and the thermal state as a function of time for various initial states, from $|Z+⟩$ ($\beta =0.7275$) to $|Y+⟩$ ($\beta =0$), in the left-hand plot, and from $|X+⟩$ ($\beta =-0.7180$) to $|Y+⟩$ in the right-hand plot. Insets show the initial states on the Bloch sphere.