Refinements of the Eigenstate Thermalization Hypothesis under Local Rotational Invariance via Free Probability
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| Format: | Preprint |
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2025
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| _version_ | 1866917111724507136 |
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| author | Vallini, Elisa Foini, Laura Pappalardi, Silvia |
| author_facet | Vallini, Elisa Foini, Laura Pappalardi, Silvia |
| contents | The Eigenstate Thermalization Hypothesis (ETH) was developed as a framework for understanding how the principles of statistical mechanics emerge in the long-time limit of isolated quantum many-body systems. Since then, ETH has shifted the attention towards the study of matrix elements of physical observables in the energy eigenbasis. In this work, we revisit recent developments leading to the formulation of full ETH, a generalization of the original ETH ansatz that accounts for multi-point correlation functions. Using tools from free probability, we explore 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. This approach allows us to make quantitative predictions and derive an analytical characterization of subleading corrections to matrix-element correlations, thereby refining the ETH ansatz. Moreover, our 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. We validate our analytical predictions through comparison with numerical simulations in non-integrable Floquet systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_23217 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Refinements of the Eigenstate Thermalization Hypothesis under Local Rotational Invariance via Free Probability Vallini, Elisa Foini, Laura Pappalardi, Silvia Statistical Mechanics High Energy Physics - Theory Quantum Physics The Eigenstate Thermalization Hypothesis (ETH) was developed as a framework for understanding how the principles of statistical mechanics emerge in the long-time limit of isolated quantum many-body systems. Since then, ETH has shifted the attention towards the study of matrix elements of physical observables in the energy eigenbasis. In this work, we revisit recent developments leading to the formulation of full ETH, a generalization of the original ETH ansatz that accounts for multi-point correlation functions. Using tools from free probability, we explore 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. This approach allows us to make quantitative predictions and derive an analytical characterization of subleading corrections to matrix-element correlations, thereby refining the ETH ansatz. Moreover, our 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. We validate our analytical predictions through comparison with numerical simulations in non-integrable Floquet systems. |
| title | Refinements of the Eigenstate Thermalization Hypothesis under Local Rotational Invariance via Free Probability |
| topic | Statistical Mechanics High Energy Physics - Theory Quantum Physics |
| url | https://arxiv.org/abs/2511.23217 |