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Non-Hermitian diluted banded random matrices: Scaling of eigenfunction and spectral properties
M Hernández-Sánchez1, G Tapia-Labra1, J A Méndez-Bermúdez1,2
1Instituto de Física, <a href="https://ror.org/03p2z7827">Benemérita Universidad Autónoma de Puebla</a>, Puebla 72570, Mexico.
Physical Review. E
|November 20, 2024
Summary
We introduce the non-Hermitian diluted banded random matrix ensemble. Its eigenfunction and spectral properties exhibit scaling behavior dependent on matrix parameters, revealing a universal localization length law.
Area of Science:
- Statistical Physics
- Quantum Chaos
- Random Matrix Theory
Background:
- Random matrix theory is crucial for understanding complex quantum systems.
- Non-Hermitian matrices are increasingly relevant for open quantum systems and non-reciprocal phenomena.
Purpose of the Study:
- Introduce and analyze the non-Hermitian diluted banded random matrix (nHdBRM) ensemble.
- Investigate the scaling properties of eigenfunctions and spectra in this sparse matrix ensemble.
- Compare results with the Hermitian counterpart.
Main Methods:
- Definition of the nHdBRM ensemble with specific sparsity and bandwidth parameters.
- Extensive numerical simulations to compute spectral and eigenfunction properties.
- Analysis of scaling behavior with a defined parameter x.
Main Results:
- Eigenfunction and spectral properties of nHdBRM scale with parameter x = γ[(bα)²/N]δ.
- A universal scaling law for the normalized localization length β = x/(1+x) is identified.
- Distinct properties are observed compared to the Hermitian ensemble.
Conclusions:
- The nHdBRM ensemble provides a new framework for studying sparse non-Hermitian systems.
- The discovered scaling laws offer insights into localization phenomena in disordered systems.
- This work bridges concepts from random matrix theory and condensed matter physics.
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