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Rationally independent free fermions with local hopping.

Jonathon Riddell1, Bruno Bertini1

  • 1University of Nottingham, University of Nottingham, University of Birmingham, School of Physics and Astronomy, Edgbaston, Birmingham B15 2TT, United Kingdom; School of Physics and Astronomy, Nottingham NG7 2RD, United Kingdom; and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, Nottingham NG7 2RD, United Kingdom.

Physical Review. E
|February 7, 2025
PubMed
Summary
This summary is machine-generated.

We introduce rationally independent free fermions, which lack spectral degeneracies and exhibit generic relaxation properties. Minimal models are constructed, and their spectral statistics are analyzed, revealing they do not follow Poisson statistics.

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Area of Science:

  • Condensed Matter Physics
  • Quantum Mechanics
  • Statistical Physics

Background:

  • Rationally independent free fermions possess unique properties, including non-degenerate many-body spectra and relaxation behaviors akin to generic systems.
  • Understanding spectral statistics in non-interacting systems is crucial for characterizing their quantum behavior and thermalization properties.

Purpose of the Study:

  • To construct minimal examples of rationally independent free fermion models for any given system size in one dimension.
  • To investigate the many-body spectral statistics of these models and compare them with theoretical predictions, particularly Poisson statistics.

Main Methods:

  • Utilizing number theory results to identify minimal rationally independent free fermion models.
  • Analyzing free fermion models with chemical potential and hopping terms involving divisors of system size and incommensurate complex amplitudes.
  • Examining spectral statistics using local probes, such as the ratio of consecutive level spacings.
  • Analyzing the moments of the spectral form factor to definitively characterize the statistics.

Main Results:

  • Minimal rationally independent free fermion models were constructed for all one-dimensional system sizes.
  • Local probes suggest a resemblance to Poisson statistics due to the non-degenerate spectrum.
  • A rigorous analysis of the spectral form factor moments demonstrates that these models never exhibit true Poisson statistics.

Conclusions:

  • Rationally independent free fermions provide a unique platform for studying quantum systems with non-generic properties.
  • While local spectral properties may mimic Poisson statistics, a global analysis reveals deviations, highlighting the complexity of spectral statistics in non-interacting systems.