Strong and weak thermalization of infinite non-integrable quantum systems

TL;DR

The study demonstrates that thermalization in an infinite non-integrable quantum spin chain is richer than in classical systems, exhibiting strong, weak, and non-thermalizing regimes depending on the initial state and Hamiltonian parameters. Using a folding-based matrix-product-state approach, the authors track three-site reduced density matrices and compare them to thermal ensembles at fixed energy, revealing distinct relaxation behaviors: instantaneous convergence (strong), slow averaging convergence (weak), and persistent non-thermal behavior (no thermalization) for certain states. Importantly, these regimes arise without fine-tuning and even occur where the spectrum appears chaotic, highlighting a nuanced quantum relaxation landscape. The results underscore that quantum memory of initial conditions can persist longer than classical predictions, with implications for experiments probing out-of-equilibrium dynamics in isolated quantum systems.

Abstract

When a non-integrable system evolves out of equilibrium for a long time, local observables are expected to attain stationary expectation values, independent of the details of the initial state. However, intriguing experimental results with ultracold gases have shown no thermalization in non-integrable settings, triggering an intense theoretical effort to decide the question. Here we show that the phenomenology of thermalization in a quantum system is much richer than its classical counterpart. Using a new numerical technique, we identify two distinct thermalization 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, happens when local expectation values converge to the thermal ones 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.

Figures (21)

• Figure 1: Strong thermalization: initial state $|Y+\rangle$. The main plot shows the distance between the three-body reduced density matrix at instant $t$ and the corresponding thermal state ($\beta=0$). The error bars show the difference between the result with the largest bond dimension ($D=120$) and with a lower one ($D=60$), which gives us an estimate of the truncation error. The right inset shows the distance between the time-averaged reduced density matrix and the thermal state. We superimpose a fit to a curve decaying like $b/\sqrt{t}$ for long times (solid black line). The left inset shows the difference between the thermal expectation values and the time dependent single body observables $\langle \sigma_x\rangle$ (blue solid line), $\langle \sigma_y\rangle$ (dashed green) and $\langle \sigma_z\rangle$ (dash-dotted black line). Convergence to the thermal state is observed in all three plots.
• Figure 2: Weak thermalization: initial state $|Z+\rangle$. The distance between the reduced density matrix for three sites and the thermal state of the same energy ($\beta=0.7275$) oscillates strongly with time. The right inset shows the distance between the time averaged reduced density matrix and the thermal ensemble. The superimposed solid line, fit to a curve which behaves as $b/\sqrt{t}$ for long times, shows that the behavior is compatible with a slow convergence only in average. The left inset shows the oscillations of the individual single-body time dependent expectation values around the thermal ones. No sign of damping is observed for the longest times ($t\sim 18$) we have simulated. These results were obtained with bond dimension $D=240$, and the error bars show the difference to $D=120$ results.
• Figure 3: No thermalization observed: initial state $|X+\rangle$. The plot shows the time dependence of the distance between the evolved reduced density matrix for three sites and the thermal state ($\beta=-0.7180$). No thermalization is observed in the instantaneous density matrix, nor in the time averaged one (right inset). On the latter, we superimpose a fit of the computed values to a time dependent function, which asymptotically tends to a constant ($0.03$). The plotted results correspond to a bond dimension $D=240$, while the distance to the results with $D=120$ is shown as error bars. In the left inset we plot, for the observables that determine the one site reduced density matrix, $\langle \sigma_x\rangle$ (blue solid line), $\langle \sigma_y\rangle$ (dashed green) and $\langle \sigma_z\rangle$ (dash-dotted black), the difference with respect to the thermal values. We observe that $\langle\sigma_x\rangle$ is the responsible for the lack of thermalization, while all the other expectation values converge to the thermal averages. Studying the evolution of only a few local expectation values may not suffice then to detect the nonthermalization.
• Figure 4: Distance between the averaged 3-body reduced density matrix and the thermal state as a function of time for various product initial states, from $|Z+\rangle$ ($\beta=0.7275$) to $|Y+\rangle$ ($\beta=0$),as indicated on the Bloch sphere on the left.
• Figure 5: Level spacing distribution of the unfolded spectrum mehta for the Hamiltonian \ref{['eq:HIsingPar']} in a finite ($L=14$) system, for different Hamiltonian parameters. the superimposed curves show a Poissonian (dashed green) and Wigner (solid red) distribution, characterizing integrable and chaotic systems, respectively.