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Eigenstate Thermalization Hypothesis and Free Probability
Silvia Pappalardi1, Laura Foini2, Jorge Kurchan1
1Laboratoire de Physique de l'École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France.
Physical Review Letters
|November 4, 2022
Summary
We uncovered a link between quantum thermalization and free probability theory. This connection simplifies understanding complex correlations in quantum systems using free cumulants.
Area of Science:
- Quantum physics
- Statistical mechanics
Background:
- Quantum thermalization is explained by the eigenstate thermalization hypothesis (ETH).
- ETH describes correlations of matrix elements using typicality arguments and local rotations.
Purpose of the Study:
- To explore the relationship between ETH and free probability theory.
- To provide a mathematical framework for understanding quantum thermalization correlations.
Main Methods:
- Applied free probability theory to thermal ensembles and energy shells.
- Utilized invariance under local rotations of nearby energy levels.
- Derived an explicit formula for free cumulants.
Main Results:
- Established a direct connection between ETH and free probability theory.
- Demonstrated that higher-order correlation functions decompose into free cumulants.
- Showed that local functions of ETH operators also satisfy ETH.
Conclusions:
- Free probability theory offers a powerful tool for analyzing quantum thermalization.
- The study expands the scope of free probability and its relevance to quantum mechanics.
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