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arXiv:2603.18394

Weighted Time Averages and Weak Convergence to Equilibrium in Quantum Integrable Systems

Xinyu Liu

Published Mar 19, 2026• 0 citations
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Abstract

This paper establishes a natural quantum counterpart of weak equilibration for statistical ensembles in integrable systems. For quantum systems with pure point spectrum, single-time expectation values under unitary evolution are typically quasiperiodic, and hence generally do not admit a pointwise limit as $t\to\infty$. To overcome this difficulty, we introduce a weighted time-averaging procedure and prove that the resulting averaged dynamics converge to the diagonal (dephased) equilibrium state. We further illustrate and validate the theoretical result through a three-spin quantum integrable model.

DOI
10.48550/arxiv.2603.18394

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This paper establishes a natural quantum counterpart of weak equilibration for statistical ensembles in integrable systems. For quantum systems with pure point spectrum, single-time expectation values under unitary evolution are typically quasiperiodic, and hence generally do not admit a pointwise limit as t→∞t\to\inftyt→∞. To overcome this difficulty, we introduce a weighted time-averaging procedure and prove that the resulting averaged dynamics converge to the diagonal (dephased) equilibrium state. We further illustrate and validate the theoretical result through a three-spin quantum integrable model.
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.
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