Pseudosymmetric random matrices: Semi-Poisson and sub-Wigner statistics.

Area of Science:

• Mathematics
• Physics
• Quantum Mechanics

Background:

• Real nonsymmetric matrices can exhibit real or complex conjugate eigenvalues.
• Pseudosymmetric matrices are defined by the condition ηMη^{-1}=M^{t}, with metric η potentially dependent on matrix elements.
• Understanding eigenvalue distributions is crucial in various physical systems.

Purpose of the Study:

• To construct and analyze ensembles of large pseudosymmetric matrices.
• To investigate the nearest level spacing distributions (NLSDs) and eigenvalue distributions of these ensembles.
• To compare findings with known distributions in physical phenomena like Anderson transitions.

Main Methods:

• Construction of large ensembles of pseudosymmetric n×n matrices using random number elements.
• Numerical calculation of nearest level spacing distributions (NLSDs).
• Fitting eigenvalue distributions to a proposed hyperbolic tangent model.

Main Results:

• Conjectured NLSDs for these ensembles follow a sub-Wigner form: p_{abc}(s)=ase^{-bs^{c}} (0
• Eigenvalue distributions were found to fit the model D(ε)=A[tanh{(ε+B)/C}-tanh{(ε-B)/C}].
• The semi-Poisson distribution (c=1) closely matches a derived form for 2x2 pseudosymmetric matrices.

Conclusions:

• Pseudosymmetric matrices exhibit specific sub-Wigner NLSDs and eigenvalue distributions.
• These distributions have implications for understanding phenomena like Anderson metal-insulator transitions and topological transitions.
• The study provides insights into eigenvalue behavior in parity-time (PT)-symmetric systems.