Orateur
Description
The Eigenstate Thermalization Hypothesis (ETH) provides the standard framework for understanding thermalization in quantum chaotic many-body systems through the statistical properties of matrix elements of local observables. In contrast, the corresponding statistical description of matrix elements in integrable systems, where an extensive number of conserved charges strongly constrains the dynamics, remains far less understood. Developing such a description is essential for studying the mechanism of generalized thermalization in integrable models. In this work, we develop a multiscale framework for analyzing the statistics of off-diagonal matrix elements in integrable systems. Going beyond the conventional microcanonical ensemble, we introduce a scaling parameter that controls the magnitude of fluctuations of the conserved charges, thereby allowing us to systematically interpolate between finite microcanonical shells and ensembles with a much broader scale of fluctuations. We implement this program in a minimal integrable model that admits both efficient numerical calculations for system sizes well beyond the current state of the art and explicit analytical treatment. We show that the statistical distributions of off-diagonal matrix elements exhibit a rich dependence on the fluctuation scale, revealing how the hierarchy of conserved charges governs the structure of matrix elements. Our results provide a new perspective on the role of integrability in the statistics of matrix elements of local observables and establish a multiscale extension of the ETH paradigm for integrable quantum systems.