Institut für Theoretische Physik: Eigenstate Correlations Characterized by Free Probability Tools

Researchers at the Institut für Theoretische Physik, Universität zu Köln, are using tools from free probability to refine the Eigenstate Thermalization Hypothesis, a key framework for understanding thermalization in isolated quantum systems. Their work builds upon the Bohigas, Giannoni, Schmidt conjecture, which established that quantum chaotic systems exhibit eigenvalue statistics consistent with random matrix theory, and accounts for the implications of local rotational invariance, a property that emerges from the statistical invariance of observables under random basis transformations induced by small perturbations of the Hamiltonian. The team establishes a direct link between a property arising from random changes to the Hamiltonian and the statistical behavior of matrix elements. Their analysis links the statistical properties of matrix elements under random basis changes to the empirical averages over energy windows. Validating analytical predictions through comparison with numerical simulations in non-integrable Floquet systems, the researchers demonstrate a crucial link between mathematical models and observable phenomena.

Eigenstate Thermalization Hypothesis and Quantum Chaos Origins

Refinements of the Eigenstate Thermalization Hypothesis (ETH) are now using tools from free probability to probe the origins of quantum chaos, moving beyond initial formulations to account for complex multi-point correlation functions. Researchers Elisa Vallini, Laura Foini, and Silvia Pappalardi, affiliated with the Universität zu Köln and the CNRS, are using this mathematical framework to understand how statistical mechanics arises in isolated quantum systems. Their work centers on a property that emerges from the statistical invariance of observables under random basis transformations induced by small perturbations of the Hamiltonian. This approach allows for quantitative predictions and a detailed analytical characterization of subtle corrections to matrix-element correlations, refining the original ETH ansatz. The team’s analysis links the statistical properties of matrix elements under random basis changes to the empirical averages over energy windows commonly used when studying quantum ensembles.

Randomness emerges as the underlying mechanism; subject to the constraint of preserving a minimal structure in energy, chaotic eigenstates are as random as possible, encoding the microcanonical ensemble. This builds on foundations laid by the Bohigas, Giannoni, Schmidt conjecture, which established that chaotic systems exhibit eigenvalue statistics consistent with random matrix theory. Vallini, Foini, and Pappalardi explain that “the chaotic eigenstates are otherwise as random as possible,” a principle that underpins their investigation into the subleading corrections to the ETH framework and the role of local free cumulants.

Full ETH Formulation for Multi-Point Correlations

Researchers are increasingly employing sophisticated mathematical tools to refine the Eigenstate Thermalization Hypothesis (ETH), moving beyond its original formulation to better understand complex quantum systems. Central to this advancement is the exploration of how different parts of a system relate to each other over time. Elisa Vallini of the Universität zu Köln, Laura Foini of the CNRS, and Silvia Pappalardi, also of the Universität zu Köln, are using tools from free probability to analyze these correlations with greater precision. Their analysis links the statistical properties of matrix elements under random basis changes to the empirical averages over energy windows. By formulating a “full ETH,” they’ve developed an expression for products of matrix elements, accounting for subleading corrections and providing a more detailed analytical characterization of quantum behavior.

The team demonstrates that this invariance directly links the statistical properties of these matrix elements to the empirical averages over energy windows, refining the ETH ansatz. The work extends the original ETH framework by studying higher-order correlations, crucial for understanding quantities like out-of-time-order correlators in diverse fields from condensed matter physics to high-energy physics.

Berry’s Random Wave Conjecture for Chaotic Systems

Conventional wisdom suggests chaotic systems are inherently unpredictable, yet a deeper understanding reveals an underlying order dictated by randomness. The team’s work focuses on how matrix elements of physical observables behave within the energy eigenbasis, building on the original ETH ansatz and using tools from free probability. Their analysis links the statistical properties of matrix elements under random basis changes to the empirical averages over energy windows. Randomness emerges as the underlying mechanism; subject to the constraint of preserving a minimal structure in energy, chaotic eigenstates are otherwise as random as possible. This property emerges from the statistical invariance of observables under random basis transformations induced by small perturbations of the Hamiltonian, providing a more refined understanding of chaotic many-body systems.

Their work focuses on the implications of local rotational invariance, a property that emerges from the statistical invariance of observables under random basis transformations induced by small perturbations of the Hamiltonian. their analysis links the statistical properties of matrix elements under random basis changes to the empirical averages over energy windows that are usually considered when dealing with a single instance of the ensemble.

Their analysis clarifies how the full ETH emerges from this local rotational invariance, and provides a quantitative characterization of corrections to existing models. Refining our understanding of complex quantum systems increasingly relies on bridging analytical predictions with rigorous computational testing, a necessity underscored by recent work exploring the Eigenstate Thermalization Hypothesis (ETH). The team used tools from free probability to explore the implications of a property that emerges from the statistical invariance of observables under random basis transformations induced by small perturbations of the Hamiltonian.