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Eigenstate Thermalization, Random Matrix Theory, and Behemoths.

Ivan M Khaymovich1, Masudul Haque1,2, Paul A McClarty1

  • 1Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany.

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We introduce "behemoths," novel nonlocal operators, to explore quantum statistical mechanics. These operators reveal insights into the eigenstate thermalization hypothesis (ETH) for complex systems.

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Area of Science:

  • Quantum Statistical Mechanics
  • Quantum Chaos
  • Many-Body Physics

Background:

  • The Eigenstate Thermalization Hypothesis (ETH) is fundamental to quantum statistical mechanics.
  • Understanding ETH for nonlocal operators remains an open challenge.

Purpose of the Study:

  • To investigate the validity of ETH for nonlocal operators.
  • To construct novel nonlocal operators as building blocks for analysis.

Main Methods:

  • Construction of highly nonlocal operators, termed "behemoths," with singular distributions (width w∼D⁻¹).
  • Utilizing an analogy with random matrix theory for operator construction.
  • Comparison with numerical simulations of nonintegrable many-body systems.

Main Results:

  • Behemoths allow construction of local operators consistent with ETH (Gaussian distribution, w∼D⁻¹/²).
  • Extrapolation suggests sub-ETH behavior for typical nonlocal operators (w∼D⁻δ, 0<δ<1/2).
  • Demonstrated striking agreement between the operator construction and numerical simulations.

Conclusions:

  • The study provides a framework for analyzing ETH in the context of nonlocal operators.
  • The findings suggest deviations from ETH for certain nonlocal operators in many-body systems.
  • The approach offers new tools for studying quantum chaos and thermalization.