Wigner-Dyson statistics for a class of integrable models.

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.
• 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.