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Eigenstate Thermalization Hypothesis for Wigner-Type Matrices
László Erdős1, Volodymyr Riabov1
1Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria.
Communications in Mathematical Physics
|November 11, 2024
Summary
We prove the Eigenstate Thermalization Hypothesis for Wigner-type matrices, demonstrating its robustness across diverse random matrix ensembles. This confirms thermalization in quantum systems with complex structures, even with vanishing entries.
Area of Science:
- Quantum mechanics
- Random matrix theory
- Statistical physics
Background:
- The Eigenstate Thermalization Hypothesis (ETH) is a cornerstone in understanding how isolated quantum systems reach thermal equilibrium.
- Previous studies often focused on specific random matrix ensembles or simpler systems, leaving broader applicability uncertain.
- Investigating ETH in Wigner-type matrices is crucial for understanding thermalization in complex quantum systems.
Purpose of the Study:
- To rigorously prove the Eigenstate Thermalization Hypothesis for a general class of Wigner-type random matrices.
- To establish optimal control over fluctuations for observables of arbitrary rank within the bulk spectrum.
- To demonstrate the robustness of ETH under very general conditions, including those with vanishing matrix entries.
Main Methods:
- Development and application of rank-uniform optimal local laws for one and two resolvents of Wigner-type matrices.
- Analysis of observables with regular properties.
- Consideration of diverse variance profiles, including those with many vanishing entries.
Main Results:
- Rigorous proof of the Eigenstate Thermalization Hypothesis for general Wigner-type matrices in the spectral bulk.
- Optimal control on fluctuations for arbitrary rank observables is achieved.
- Demonstration that ETH holds robustly across a diverse class of random matrix ensembles.
Conclusions:
- The Eigenstate Thermalization Hypothesis is proven to be a robust phenomenon for Wigner-type matrices.
- The findings confirm ETH's validity in quantum systems with non-trivial spatial structures and under general conditions.
- This work provides a significant theoretical advancement in understanding thermalization in complex quantum systems.
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