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Related Experiment Videos

Wigner-Dyson statistics for a class of integrable models.

L Benet1, F Leyvraz, T H Seligman

  • 1Centro de Ciencias Físicas, UNAM, Apartado Postal 48-3, 62251 Cuernavaca, Morelos, Mexico.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 20, 2003
PubMed
Summary

We created quantum Hamiltonians exhibiting Gaussian orthogonal (GOE) or Gaussian unitary (GUE) statistics. These systems are integrable due to conserved boson number, offering a novel construction method.

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Area of Science:

  • Quantum mechanics
  • Statistical physics
  • Many-body systems

Background:

  • Ensembles of Hamiltonians are crucial for understanding complex quantum systems.
  • Gaussian ensembles (GOE, GUE) describe random matrix properties.
  • Integrable systems possess conserved quantities, simplifying their analysis.

Purpose of the Study:

  • To construct an ensemble of quantum Hamiltonians with specific statistical properties.
  • To demonstrate that these Hamiltonians can be integrable.
  • To explore the connection between integrability and random matrix theory.

Main Methods:

  • Constructing second-quantized Hamiltonians with two bosonic degrees of freedom.
  • Employing reverse engineering based on the classical limit of interacting bosons.

Related Experiment Videos

  • Utilizing Heisenberg's association of boson operators to action-angle variables.
  • Selecting n-body random interactions and degenerate energy levels.

    • Main Results:

    • The constructed ensemble members exhibit Gaussian orthogonal ensemble (GOE) or Gaussian unitary ensemble (GUE) statistics with probability one.
    • These Hamiltonians possess an additional integral of motion: the boson number, rendering them integrable.
    • The method connects random matrix theory with integrable systems through a novel construction.

    Conclusions:

    • A method for constructing integrable quantum Hamiltonians with GOE/GUE statistics is presented.
    • The boson number serves as a key integral of motion in these systems.
    • This work bridges concepts from random matrix theory and the study of integrable systems.