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Exact probability distributions of complex spacing ratios in non-Hermitian random matrices

This paper derives exact finite- $N$ complex spacing ratio distributions for non-Hermitian Gaussian ensembles in classes AI $^{\dag}$ and AII $^{\dag}$ , providing explicit algebraic expressions for class AII $^{\dag}$ and a normalized integral representation with asymptotic analysis for class AI $^{\dag}$ , all validated by numerical diagonalization.

Original authors: Kohei Kawabata

Published 2026-09-29

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Kohei Kawabata

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of quantum physics, some systems are so intricate that their inner workings seem impenetrable. To make sense of them, scientists often turn to a statistical approach, treating the energy levels of a complex system not as a rigid list of numbers, but as a pattern emerging from a crowd. This method, known as random matrix theory, has long been the standard for understanding closed systems where energy is conserved. However, the real world is rarely closed. Most quantum systems we encounter are open, constantly exchanging energy and matter with their surroundings. In these open environments, the mathematical tools used for closed systems break down because the energy values are no longer simple, real numbers but complex values that exist on a two-dimensional plane. Understanding how these complex values arrange themselves is crucial for diagnosing whether a system behaves in a predictable, orderly way or in a chaotic, unpredictable manner.

A powerful way to probe this arrangement is by looking at the spacing between neighboring energy values. Instead of measuring the distance between two points directly, which can be distorted by how crowded the area is, researchers look at the ratio of one spacing to the next. Imagine standing in a forest and measuring the distance to the nearest tree, then the distance to the second-nearest tree. The ratio of these two distances tells you about the local structure of the forest without needing to know the overall density of trees. In the complex world of open quantum systems, this ratio is not just a number but a direction and a length combined, forming a complex number that reveals how the energy levels repel or attract one another.

In a recent study, a researcher at the University of Tokyo has mapped out exactly how these complex spacing ratios behave in specific types of open quantum systems. The work focuses on two distinct categories of systems that possess a special kind of symmetry, often described as time-reversal symmetry but defined in a way that applies to open systems. These categories, known as class AI-dagger and class AII-dagger, represent different ways that the system's rules interact with the complex nature of its energy values. While the general behavior of these systems was known to be different from standard ones, the precise mathematical shape of their spacing ratios had remained a mystery, especially for systems of a finite, manageable size. The researcher has now derived exact formulas that describe these distributions, providing a clear, quantitative picture of how energy levels organize themselves in these specific open environments.

The study begins by establishing a baseline using a well-understood category of systems, where the energy values are unrestricted complex numbers. For this category, the researcher confirms that the spacing ratios follow a predictable pattern that can be calculated exactly for any number of energy levels. This serves as a control, showing how the system behaves without the specific symmetries that define the more complex cases. The real breakthrough comes when the researcher turns to the two symmetric classes. For the class known as AII-dagger, which involves matrices with a specific self-dual structure, the researcher derived a complete algebraic expression. This formula works for any number of energy levels, allowing for precise predictions of the spacing ratios. The researcher tested this formula for small systems with three, four, five, and six energy levels, calculating the exact average distances and angles between them. These calculations revealed that the symmetry in this class leads to a unique pattern of repulsion between energy levels, one that is stronger than in the standard case and changes in a non-monotonic way as the system grows larger.

For the other symmetric class, AI-dagger, the situation is more complicated. The mathematical description of these systems involves an integral over extra variables that cannot be easily simplified into a neat algebraic formula. Despite this difficulty, the researcher managed to derive an exact integral representation for a system with three energy levels. This formula, while more complex to compute, allowed for a deep analysis of the system's behavior at the extremes. The researcher found that when energy levels get very close to each other, the probability of finding them there does not drop off smoothly. Instead, it is modified by a logarithmic factor, a subtle but significant correction that changes the nature of the repulsion. Furthermore, the researcher discovered that the distribution of angles between the spacing ratios contains a non-smooth feature. Unlike other systems where the angular distribution is perfectly smooth, this class exhibits a sharp kink in its mathematical description, indicating a fundamental difference in how the energy levels arrange themselves in space.

To ensure these theoretical findings were correct, the researcher compared the derived formulas against direct numerical simulations. In these simulations, thousands of random matrices were generated and their energy values calculated to see what the spacing ratios actually looked like. The results from the simulations matched the theoretical predictions with high precision, confirming that the derived formulas accurately capture the physics of these systems. The study also looked at how these patterns evolve as the number of energy levels increases. In the AII-dagger class, the researcher observed that the average spacing ratio does not simply settle into a steady value as the system grows; instead, it rises and then falls before stabilizing, a behavior that highlights the importance of studying finite-sized systems rather than just assuming they behave like infinitely large ones.

The implications of this work extend beyond the specific formulas derived. By providing exact distributions for finite systems, the researcher has created a set of benchmarks that other scientists can use to test their own models of open quantum systems. These benchmarks are essential for distinguishing between universal behaviors that apply to all systems of a certain type and specific effects that depend on the details of a particular model. The discovery of the logarithmic correction and the non-smooth angular density in the AI-dagger class suggests that these systems possess unique structural properties that were previously hidden. The work also clarifies the limitations of current methods, showing that while some classes of systems can be described by simple algebraic formulas, others require more sophisticated integral representations.

Ultimately, this research bridges the gap between abstract mathematical theory and the concrete behavior of complex quantum systems. It demonstrates that even in the chaotic realm of open quantum systems, there are precise, predictable patterns waiting to be uncovered. The ability to calculate these patterns exactly, rather than relying on approximations, gives scientists a new tool for diagnosing the nature of quantum chaos and non-integrability. As the field moves toward understanding larger and more complex open systems, these exact results for smaller systems serve as a critical foundation, ensuring that the interpretations of experimental data are built on solid theoretical ground. The study leaves open the question of how these patterns behave in the limit of infinitely large systems, a challenge that remains for future investigation, but for now, it has provided a definitive map of the terrain for systems of finite size.

Technical Summary: Exact Probability Distributions of Complex Spacing Ratios in Non-Hermitian Random Matrices

Problem Statement
While Hermitian random matrices have long provided a statistical framework for energy spectra in closed quantum systems, the characterization of open quantum systems requires the study of non-Hermitian random matrices. In these systems, local spectral correlations are governed by an enriched symmetry classification, specifically the non-Hermitian extensions of Dyson's threefold way: classes A, AI $^\dagger$ , and AII $^\dagger$ . A robust diagnostic for spectral correlations and nonintegrability in these systems is the complex spacing ratio, defined as the complex displacement from a reference eigenvalue to its nearest neighbor divided by the displacement to its next-to-nearest neighbor.

Despite the utility of this statistic, analytical results for complex spacing ratio distributions in non-Hermitian classes AI $^\dagger$ (complex symmetric matrices) and AII $^\dagger$ (complex self-dual matrices) have been incomplete. Existing results for class A (complex Ginibre) are limited to specific finite- $N$ cases or rely on surmises that deviate significantly from large- $N$ behavior for small matrix dimensions. Furthermore, the joint eigenvalue probability densities for classes AI $^\dagger$ and AII $^\dagger$ involve irreducible many-eigenvalue interactions that cannot be reduced to simple pairwise Coulomb-gas structures, complicating the derivation of spacing distributions. This work addresses the gap by deriving exact finite- $N$ complex-spacing-ratio distributions for classes AI $^\dagger$ and AII $^\dagger$ .

Methodology
The author derives the distributions starting from the exact finite- $N$ joint eigenvalue probability densities ( $\rho_N$ ) for Gaussian ensembles in classes AI $^\dagger$ and AII $^\dagger$ , recently obtained in Ref. [52] via connections to the Calogero model.

Origin-Conditioned Formulation: To simplify the problem and approximate universal bulk statistics, the author conditions on the presence of a reference eigenvalue at the spectral origin ( $z_1 = 0$ ). They retain the neighbor-ordering constraints ( $|z_2| < |z_3| < |z_n|$ for $n \geq 4$ ) to define the complex spacing ratio $\eta = z_2/z_3 = r e^{i\theta}$ .
Integration Strategy: The probability density $p_N^{(0)}(\eta)$ is formulated as an integral over the remaining eigenvalues, utilizing the delta function to enforce the ratio definition and Heaviside step functions to enforce neighbor ordering.
Class AII $^\dagger$ (Complex Self-Dual): For this class, the joint density is a finite polynomial involving the Vandermonde determinant and a many-body polynomial $R_N$ . The author performs explicit integrations over the remaining eigenvalues. For arbitrary $N$ , they derive an exact algebraic expression by expanding the integrand into monomials and utilizing Gaussian integral identities, resulting in a sum of terms involving factorials and polynomials in $u=|\eta|^2$ and $v=|1-\eta|^2$ .
Class AI $^\dagger$ (Complex Symmetric): The joint density for this class contains a noncompact orbital integral involving the complete elliptic integral of the first kind, $K$ . For $N=3$ , the author analytically reduces the multi-dimensional integral to a normalized one-dimensional integral representation. For generic $N$ , they employ a Monte Carlo integration method based on the exact joint density measure (without generating and diagonalizing random matrices) to evaluate the distributions.
Validation: Analytical results are cross-verified against independent numerical diagonalization of non-Hermitian random matrices and, for class AI $^\dagger$ , against Monte Carlo sampling of the exact joint density.

Key Contributions and Results

Class AII $^\dagger$ :
- An exact algebraic expression for the complex-spacing-ratio distribution is derived for arbitrary finite $N$ .
- Explicit distributions and representative moments (e.g., $\langle r \rangle$ , $\langle \cos \theta \rangle$ ) are calculated for $N = 3, 4, 5, 6$ .
- The results reveal a nonmonotonic finite-size behavior: the mean ratio $\langle r \rangle$ initially increases with $N$ before decreasing toward the large- $N$ limit, contrasting with the monotonic behavior observed in classes A and AI $^\dagger$ .
- For $N=3$ , the distribution exhibits enhanced level repulsion compared to class A, indicated by a larger mean ratio and more negative $\langle \cos \theta \rangle$ .
Class AI $^\dagger$ :
- For $N=3$ , an exact integral representation involving the complete elliptic integral of the first kind is derived.
- Asymptotic Analysis: The author identifies a logarithmic correction to the cubic level repulsion ( $p(r) \sim r^3 \log(1/r)$ ) and a nonanalytic contribution to the angular density ( $|\theta|^3$ term).
- Angular Moments: Unlike classes A and AII $^\dagger$ , where higher-order angular moments vanish for sufficiently large $n$ , the moments $\langle \cos n\theta \rangle$ in class AI $^\dagger$ decay algebraically as $n^{-4}$ and remain nonvanishing for arbitrarily large $n$ . This persistence of higher-order harmonics is identified as a hallmark of the AI $^\dagger$ symmetry class.
- For $N > 3$ , Monte Carlo estimates based on the exact joint density confirm consistency with direct numerical diagonalization.
Comparison with Class A:
- The paper provides a unified comparison across the non-Hermitian threefold way. While all classes share the level-repulsion factor $|\eta|^2 |1-\eta|^2$ , the specific functional forms and finite-size behaviors differ significantly.
- The results for class A are reviewed and extended to arbitrary $N$ using a determinant formulation, serving as a baseline for comparison.

Significance
The paper establishes finite- $N$ benchmarks for complex-spacing-ratio distributions across the non-Hermitian threefold way. By providing exact algebraic expressions for class AII $^\dagger$ and integral representations for class AI $^\dagger$ , the work moves beyond approximate surmises and captures finite-size effects that are crucial for small non-Hermitian random matrices. The identification of distinct asymptotic behaviors—specifically the logarithmic corrections and nonanalytic angular densities in class AI $^\dagger$ —highlights the unique spectral correlations induced by time-reversal symmetry $^\dagger$ . These results serve as a foundation for future controlled asymptotic analyses of the large- $N$ limit and offer precise tools for diagnosing nonintegrability and spectral correlations in open quantum systems.

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