[Paper Review] Non-Hermiticity and Universality

Pragya Shukla|arXiv (Cornell University)|May 1, 2001

Quantum Mechanics and Non-Hermitian Physics15 citations

TL;DR

This paper establishes a mapping between the eigenvalue statistics of non-Hermitian random matrices and a two-dimensional Calogero-Sutherland (CS) Hamiltonian, revealing universal spectral correlations governed by inverse-square and three-body interactions. The key result is that eigenvalue correlations in diverse dissipative complex systems are unified under this integrable framework, implying deep universality in non-Hermitian spectral behavior across different physical systems and complexity parameters.

ABSTRACT

. We study the statistical properties of the eigenvalues of non-Hermitian operators assoicated with the dissipative complex systems. By considering the Gaussian ensembles of such operators, a hierarchical relation between the correlators is obtained. Further the eigenvalues are found to behave like particles moving on a complex plane under 2-body (inverse square) and 3-body interactions and there seems to underlie a deep connection and universality in the spectral behaviour of different complex systems. .

Motivation & Objective

• To understand the statistical properties of eigenvalues in non-Hermitian operators arising in dissipative complex systems such as chaotic scattering and neural networks.
• To establish a connection between the eigenvalue distribution of non-Hermitian random matrices and the dynamics of a classically integrable 2D Calogero-Sutherland system.
• To demonstrate that eigenvalue correlations in diverse non-Hermitian systems are governed by a universal mechanism rooted in long-range interactions and integrability.
• To show that the evolution of eigenvalue statistics under varying complexity parameters corresponds to time evolution in the CS system, enabling unified analysis.

Proposed method

• A mapping is constructed from the eigenvalue distribution of non-Hermitian random matrices to a non-stationary state of a 2D Calogero-Sutherland Hamiltonian with inverse-square and three-body interactions.
• The method uses a parametric evolution equation for the eigenvalue distribution, expressed as a sum S involving derivatives with respect to matrix element variances and covariances.
• The eigenvalue and eigenvector derivatives under small matrix perturbations are derived using perturbation theory, enabling the transformation to particle dynamics.
• The Schrödinger-type equation ∂Ψ/∂Y = −ĤΨ is derived, where Ĥ is the 2D CS Hamiltonian with terms for kinetic energy, inverse-square two-body repulsion, three-body interactions, and harmonic confinement.
• The transformation Ψ = P / |Q_N|^{β/2} maps the eigenvalue distribution P to a wavefunction Ψ of the CS system, allowing use of known integrable system solutions.
• The joint eigenvalue distribution P is expressed as a sum over CS eigenstates, enabling the calculation of static and dynamic correlation functions via spectral decomposition.

Experimental results

Research questions

• RQ1 Can the statistical properties of eigenvalues in non-Hermitian random matrices be universally described across different dissipative systems?
• RQ2 Is there a deep connection between the spectral correlations of non-Hermitian operators and the dynamics of integrable many-body systems?
• RQ3 How do eigenvalue correlations evolve with increasing system complexity, and can this evolution be described by a single underlying Hamiltonian?
• RQ4 To what extent do the eigenvalue statistics of non-Hermitian systems depend on the specific distribution of matrix elements?

Key findings

• The eigenvalue distribution of non-Hermitian random matrices is mapped to a non-stationary state of a 2D Calogero-Sutherland Hamiltonian, revealing a universal dynamical framework.
• The eigenvalue correlations in non-Hermitian systems are governed by 2-body inverse-square and 3-body interactions, analogous to the CS model in two dimensions.
• The mapping shows that different non-Hermitian systems with the same complexity parameter Y−Y₀ exhibit identical eigenvalue statistics, indicating universality.
• The joint probability distribution P(μ,Y) for eigenvalues is expressed as a sum over CS eigenstates, enabling the calculation of static and dynamic correlation functions.
• The correspondence holds for both Gaussian and potentially non-Gaussian ensembles, as long as the matrix element distribution is smooth and the transformation to eigenvalue space is valid.
• The method allows the transfer of knowledge from the well-studied integrable CS system to the analysis of eigenvalue statistics in complex non-Hermitian systems.