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Goran Nakerst1, Masudul Haque1,2

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The eigenstate thermalization hypothesis (ETH) predicts fluctuations decrease with system size. This study finds anomalous scaling in smaller systems, deviating from ETH expectations for quantum chaotic systems.

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Area of Science:

  • Quantum physics
  • Statistical mechanics
  • Condensed matter theory

Background:

  • The eigenstate thermalization hypothesis (ETH) posits that expectation values of local observables in many-body quantum systems fluctuate minimally between energy eigenstates.
  • These fluctuations are expected to decrease with increasing system size, scaling as D^{-1/2} where D is the Hilbert space dimension, as the system approaches the thermodynamic limit.

Purpose of the Study:

  • To investigate the scaling of eigenstate expectation value fluctuations in the classical or semiclassical limit, distinct from the thermodynamic limit.
  • To examine the validity of ETH predictions in the Bose-Hubbard model across different lattice sizes and behaviors (quantum chaotic vs. mixed).

Main Methods:

  • Derivation of theoretical expressions for expected scaling of fluctuations, assuming Gaussian-distributed random components in ideal eigenstates.
  • Numerical simulations of the Bose-Hubbard model on fixed lattice topologies to observe scaling behavior.
  • Analysis of midspectrum eigenstates in larger (quantum chaotic) and smaller (mixed behavior) lattices.

Main Results:

  • For larger lattices, numerical results confirm the ETH scaling of fluctuations, aligning with the Gaussian expectation.
  • For smaller lattices, anomalous scaling exponents were observed, deviating from the standard ETH prediction.
  • Identification of potential mechanisms responsible for this deviation in smaller, non-fully chaotic systems.

Conclusions:

  • While ETH holds for larger, quantum chaotic systems, its predictions for fluctuation scaling are modified in smaller systems exhibiting mixed quantum behavior.
  • The study highlights the importance of system size and the nature of quantum chaos in determining the validity and form of ETH predictions.
  • Further investigation is warranted to fully elucidate the mechanisms behind the anomalous scaling observed in smaller lattices.