Gradual eigenvector ergodization in coupled Ginibre matrices

Abstract: Non-Hermitian random matrices provide a useful framework for understanding universal characteristics of dissipative quantum chaotic systems with loss or gain. We consider a model of two such system represented by two independent $N\times N$ complex Ginibre matrices interacting via a deterministic matrix $c{\bf 1}_N$, where $c$ is the complex coupling parameter whose magnitude $|c|$ controls the interaction strength. We characterize quantitatively how the eigenvectors of the whole system, initially localized in one of the individual subsystems for $|c|=0$, eventually spread over the full system with growing interaction strength. The resulting asymptotic formula describing such spread in the limit $N\to \infty$ is very explicit and provides a full picture of the gradual ergodization of eigenvectors as a function of the coupling parameter $|c|$ in the whole transition regime. As a by-product of our method we also compute the mean eigenvalue density for our model at the origin of the spectral bulk $z=0$ in the fully ergodic regime, when the coupling is scaled with the matrix size as $c=\sqrt{N}\tilde{c}$. We find that as $N\to \infty$ the limiting density at the origin vanishes beyond the critical value $|\tilde{c}|=1$, signalling of a split of the density support in the complex plane into two disjoint domains.

• The paper derives an explicit formula for the limiting joint density of eigenvector weights, capturing the transition from localized to ergodic regimes.
• It employs a Kac-Rice integral representation to analytically solve eigenvector statistics in a 2N×2N non-Hermitian block matrix with tunable coupling.
• Results reveal a critical coupling threshold where eigenstate ergodicity emerges, offering new insights into quantum chaos and non-Hermitian random matrix theory.

Gradual Eigenvector Ergodization in Coupled Ginibre Matrices

Overview and Motivation

The study presents a comprehensive and rigorous analysis of the ergodization process for eigenvectors in a model comprising two coupled $N \times N$ complex Ginibre ensembles, with particular focus on the statistical behavior of right eigenvector weights across a tunable coupling regime. The model is realized via a $2N \times 2N$ non-Hermitian block matrix with coupling parameter $c \in \mathbb{C}$, interpolating between two decoupled Ginibre systems and the fully coupled regime. The research addresses, in the large $N$ limit, the statistical transition from localized eigenvectors (inhabiting distinct subsystems) to fully ergodic eigenvectors uniformly spanning both subspaces as the coupling increases.

The work is situated within the context of quantum chaos and non-Hermitian random matrix theory (RMT), directly relevant to dissipative quantum systems where gain/loss mechanisms break Hermiticity. The results are also significant for understanding fundamental questions of the Eigenstate Thermalization Hypothesis (ETH) in non-Hermitian settings, and the universality of ergodic features under random matrix modeling.

Model Description

The core object is the $2N \times 2N$ matrix ensemble

$$X = \begin{pmatrix} G_1 & c \mathbf{1}_N \ \overline{c} \mathbf{1}_N & G_2 \end{pmatrix}, \quad c\in\mathbb{C}$$

where $G_1$, $G_2$ are independent Ginibre matrices and $\mathbf{1}_N$ is the $N \times N$ identity. For $2N \times 2N$0, the spectrum and eigenvectors are trivially those of the uncoupled subsystems, whereas finite $2N \times 2N$1 mixes the subsystems via direct coupling.

The key eigenvector statistic is the pair of squared weights (norms) $2N \times 2N$2, $2N \times 2N$3 for a normalized eigenvector $2N \times 2N$4. The central object of interest is the joint probability density (JPD) $2N \times 2N$5 at a spectral point $2N \times 2N$6, capturing the distribution of the eigenvector weights for eigenvalues near $2N \times 2N$7.

Main Results

A major analytic achievement of the paper is the derivation of an explicit formula for the limiting (as $2N \times 2N$8) JPD at $2N \times 2N$9:

$c \in \mathbb{C}$0

where $c \in \mathbb{C}$1 is the modified Bessel function of the first kind. This provides a full description of the ergodization of the eigenvector weights as a function of the coupling strength $c \in \mathbb{C}$2, interpolating smoothly from fully localized to fully ergodic regimes.

Additionally, the marginal distribution for one component $c \in \mathbb{C}$3 is

$c \in \mathbb{C}$4

which captures the progression and crossover of ergodic properties.

Figure 1: The density $c \in \mathbb{C}$5 for several values of $c \in \mathbb{C}$6, showing the evolution from a double-peak structure (localized) toward a single, symmetric, ergodic peak as the coupling increases.

The asymptotic mean eigenvalue density at the spectral origin is calculated to be $c \in \mathbb{C}$7 in the transition regime (fixed $c \in \mathbb{C}$8), and undergoes a structural transition for extensive coupling $c \in \mathbb{C}$9. In this latter case, the density vanishes at $N$0, indicating the division of the density support into disjoint domains.

Analysis of the Ergodization Transition

The limiting distributions reveal several salient features of eigenvector statistics under increasing coupling:

• Zero Coupling $N$1: The JPD reduces to two delta functions at $N$2, $N$3 and $N$4, $N$5, reflecting complete localization in subsystem 1 or 2 respectively.
• Strong Coupling $N$6: The distribution concentrates as a sharp Gaussian around $N$7, denoting full ergodicity and uniform weight across both subsystems.
• Intermediate Regime: The density $N$8 presents either two maxima away from $N$9 (weak coupling), signaling persistence of subsystem identity, or a single central maximum (stronger coupling), reflecting the loss of identity and the onset of full ergodicity. A critical coupling $2N \times 2N$0 marks the crossover where the double peak merges into a single peak.
• Spectral Density Transition: For extensive scaling $2N \times 2N$1, there is an abrupt transition at $2N \times 2N$2, linked to the splitting of the complex spectral support into two disconnected domains—a phenomenon with analogs in related non-Hermitian matrix problems.

These analytic findings quantitatively describe the crossover from subsystem-localized to globally ergodic eigenstates, and precisely characterize the loss-of-identity threshold in terms of the coupling $2N \times 2N$3, providing interpretable criteria for ergodicity in non-Hermitian coupled systems.

Methodological Advances

The derivation employs a recent approach by Fyodorov, based on a Kac-Rice-type integral representation for the joint density of eigenvalues and eigenvectors, which offers advantages over standard Girko or Efetov supersymmetry approaches. This framework enables tractable computation of finite-$2N \times 2N$4 quantities and facilitates asymptotic analysis, producing explicit and manageable expressions for the relevant densities.

A key technical achievement is the demonstration that the eigenvector ergodization described is universal in the large $2N \times 2N$5 limit and robust under generalizations, such as more generic non-Hermitian ensembles and different forms of the coupling.

Theoretical and Practical Implications

From a theoretical perspective, the results contribute to a deeper understanding of ETH, quantum chaos, and the universality of eigenstate statistics in open quantum systems. The model provides a quantitative and exactly solvable paradigm for tracking the ergodization of eigenvectors, directly relevant for quantum chaotic systems with loss/gain described by non-Hermitian RMT.

Practically, the insights inform expectations for the spread and structure of eigenstates in models of coupled dissipative systems, including potential applications in non-Hermitian extensions of the Sachdev-Ye-Kitaev model, many-body localization, and random band matrices. The identification of a clear ergodic transition controlled by a tunable parameter opens paths for further numerical and analytical studies, including the analysis of eigenvalue correlations and the fate of ergodicity in spatially structured random systems.

Connections and Outlook

The work connects directly with several themes in contemporary mathematical physics:

• Quantum Thermalization and ETH: The results extend the scope of ETH and quantum unique ergodicity notions to the non-Hermitian domain.
• Random Matrix Theory: The analytic tractability exemplifies the power of RMT tools for extracting explicit spectral and eigenfunction statistics, potentially extensible to non-Gaussian matrices.
• Supersymmetric and Kac-Rice Techniques: The adoption of the Kac-Rice-inspired approach signals new directions for non-Hermitian eigenvector theory, with generalizations conceivable for more complex or physically structured models.

Future directions include exploration of eigenvalue-eigenvector correlations in the crossover regime, application of the method to multi-block or banded random matrices (with relevance to Anderson (de)localization), and extension to settings with explicit spatial or network structure.

Conclusion

The paper provides a rigorous and highly explicit quantitative framework for understanding the ergodization of eigenvectors in coupled non-Hermitian random matrix ensembles. It identifies, for the first time, analytic forms for the gradual transition from localization to ergodicity as a function of system coupling, establishes the conditions for subsystem identity loss, and characterizes accompanying changes in the eigenvalue spectral density. These results form a critical addition to the theory of non-Hermitian quantum systems, opening new avenues for analysis and computation in both mathematical physics and quantum statistical mechanics.