non-Hermitian Random Matrix Theory Research Articles

Interactions between different birds of prey as a random point process

The two-dimensional (2D) Coulomb gas is a one-parameter family of random point processes, depending on the inverse temperature β. Based on previous work, it is proposed as a simple statistical measure to quantify the intra- and interspecies repulsion among three different highly territorial birds of prey. Using data from the area of the Teutoburger Wald over 20 years, we fit the nearest-neighbour and next-to-nearest neighbour spacing distributions between the respective nests of the goshawk, eagle owl and the previously examined common buzzard to β of the Coulomb gas. Within each species, the repulsion measured in this way deviates significantly from the Poisson process of independent points in the plane. In contrast, the repulsion amongst each of two species is found to be considerably lower and closer to Poisson. Methodologically, we investigate the influence of the terrain, of a shorter interaction range given by the 2D Yukawa interaction, and the statistical independence of the time moving average we use for the yearly ensembles of occupied nests. We also check that an artificial random displacement of the original nest positions of the order of the mean level spacing quickly destroys the repulsion measured by β > 0. A simple, approximate analytical expression for the nearest-neighbour spacing distribution derived from non-Hermitian random matrix theory proves to be very useful.

Duality in non-Hermitian random matrix theory

We consider 9 Gaussian matrix ensembles characterized by single symmetry among the 38-fold symmetry classification classes of non-Hermitian random matrices, and establish exact duality formulae of certain observables between them. Particularly, averaged products of K characteristic polynomials in an N×N matrix ensemble can be expressed in terms of another K×K matrix ensemble. Our method is to combine matrix-valued heat equations and differential identities for determinants and Pfaffians and has more possible applications.

Spectral properties of disordered interacting non-Hermitian systems

Non-hermitian systems have gained a lot of interest in recent years. However, notions of chaos and localization in such systems have not reached the same level of maturity as in the Hermitian systems. Here, we consider non-hermitian interacting disordered Hamiltonians and attempt to analyze their chaotic behavior or lack of it through the lens of the recently introduced non-hermitian analog of the spectral form factor and the complex spacing ratio. We consider three widely relevant non-hermitian models which are unique in their ways and serve as excellent platforms for such investigations. Two of the models considered are short-ranged and have different symmetries. The third model is long-ranged, whose hermitian counterpart has itself become a subject of growing interest. All these models exhibit a deep connection with the non-hermitian Random Matrix Theory of corresponding symmetry classes at relatively weak disorder. At relatively strong disorder, the models show the absence of complex eigenvalue correlation, thereby, corresponding to Poisson statistics. Our thorough analysis is expected to play a crucial role in understanding disordered open quantum systems in general.

Extended Anderson Criticality in Heavy-Tailed Neural Networks.

We investigate the emergence of complex dynamics in networks with heavy-tailed connectivity by developing a non-Hermitian random matrix theory. We uncover the existence of an extended critical regime of spatially multifractal fluctuations between the quiescent and active phases. This multifractal critical phase combines features of localization and delocalization and differs from the edge of chaos in classical networks by the appearance of universal hallmarks of Anderson criticality over an extended region in phase space. We show that the rich nonlinear response properties of the extended critical regime can account for a variety of neural dynamics such as the diversity of timescales, providing a computational advantage for persistent classification in a reservoir setting.

Dissipative quantum dynamics, phase transitions, and non-Hermitian random matrices

We explore the connections between dissipative quantum phase transitions and non-Hermitian random matrix theory. For this, we work in the framework of the dissipative Dicke model which is archetypal of symmetry-breaking phase transitions in open quantum systems. We establish that the Liouvillian describing the quantum dynamics exhibits distinct spectral features of integrable and chaotic character on the two sides of the critical point. We follow the distribution of the spacings of the complex Liouvillian eigenvalues across the critical point. In the normal and superradiant phases, the distributions are two-dimensional Poisson and that of the Ginibre unitary random matrix ensemble, respectively. Our results are corroborated by computing a recently introduced complex-plane generalization of the consecutive level-spacing ratio distribution. Our approach can be readily adapted for classifying the nature of quantum dynamics across dissipative critical points in other open quantum systems.

New universality classes of the non-Hermitian Dirac operator in QCD-like theories

In non-Hermitian random matrix theory there are three universality classes for local spectral correlations: the Ginibre class and the nonstandard classes ${\mathrm{AI}}^{\ifmmode\dagger\else\textdagger\fi{}}$ and ${\mathrm{AII}}^{\ifmmode\dagger\else\textdagger\fi{}}$. We show that the continuum Dirac operator in two-color QCD coupled to a chiral U(1) gauge field or an imaginary chiral chemical potential falls in class ${\mathrm{AI}}^{\ifmmode\dagger\else\textdagger\fi{}}$ (${\mathrm{AII}}^{\ifmmode\dagger\else\textdagger\fi{}}$) for fermions in pseudoreal (real) representations of SU(2). We introduce the corresponding chiral random matrix theories and verify our predictions in lattice simulations with staggered fermions, for which the correspondence between representation and universality class is reversed. Specifically, we compute the complex eigenvalue spacing ratios introduced recently. We also derive novel spectral sum rules.

Local microscopic behavior for 2D Coulomb gases

The study of two-dimensional Coulomb gases lies at the interface of statistical physics and non-Hermitian random matrix theory. In this paper we give a large deviation principle (LDP) for the empirical fields obtained, under the canonical Gibbs measure, by zooming around a point in the bulk of the equilibrium measure, up to the finest averaging scale \(N^{-1/2 + \varepsilon }\). The rate function is given by the sum of the “renormalized energy” of Serfaty et al. weighted by the inverse temperature, and of the specific relative entropy. We deduce a local law which quantifies the convergence of the empirical measures of the particles to the equilibrium measure, up to the finest scale.

Spectral statistics in directed complex networks and universality of the Ginibre ensemble

Spectra of the adjacency matrices of directed complex networks are analyzed by using non-Hermitian random matrix theory. Both the short-range and long-range correlations in the eigenvalues are calculated numerically for directed model complex networks and real-world networks. The results are compared with predictions of Ginibre’s ensemble. The spectral density ρ(λ), the nearest neighbor spacing distribution p(s) and the level-number variance Σ2(L) show good agreement with Ginibre’s ensemble when the adjacency matrices of directed complex networks are in the strongly non-Hermitian regime.

Gap probabilities in non-Hermitian random matrix theory

We compute the gap probability that a circle of radius r around the origin contains exactly k complex eigenvalues. Four different ensembles of random matrices are considered: the Ginibre ensembles and their chiral complex counterparts, with both complex (β=2) or quaternion real (β=4) matrix elements. For general non-Gaussian weights we give a Fredholm determinant or Pfaffian representation respectively, depending on the non-Hermiticity parameter. At maximal non-Hermiticity, that is, for rotationally invariant weights, the product of Fredholm eigenvalues for β=4 follows from the β=2 case by skipping every second factor, in contrast to the known relation for Hermitian ensembles. On additionally choosing Gaussian weights we give new explicit expressions for the Fredholm eigenvalues in the chiral case, in terms of Bessel-K and incomplete Bessel-I functions. This compares with known results for the Ginibre ensembles in terms of incomplete exponentials. Furthermore, we present an asymptotic expansion of the logarithm of the gap probability for large argument r at large N in all four ensembles, up to and including the third order linear term. We can provide strict upper and lower bounds and present numerical evidence for the conjectured values of the linear term, depending on the number of exact zero eigenvalues in the chiral ensembles. For the Ginibre ensemble at β=2, exact results were previously derived by Forrester [Phys. Lett. A 169, 21 (1992)].

Individual Complex Dirac Eigenvalue Distributions from Random Matrix Theory and Comparison to Quenched Lattice QCD with a Quark Chemical Potential

We analyze how individual eigenvalues of the QCD Dirac operator at nonzero quark chemical potential are distributed in the complex plane. Exact and approximate analytical results for both quenched and unquenched distributions are derived from non-Hermitian random matrix theory. When comparing these to quenched lattice QCD spectra close to the origin, excellent agreement is found for zero and nonzero topology at several values of the quark chemical potential. Our analytical results are also applicable to other physical systems in the same symmetry class.

Non-Hermitian random matrix theory: summation of planar diagrams, the ‘single-ring’ theorem and the disc–annulus phase transition

I review aspects of work done in collaboration with A. Zee and R. Scalettar \cite{fz1,fz2,fsz} on complex non-hermitean random matrices. I open by explaining why the bag of tools used regularly in analyzing hermitean random matrices cannot be applied directly to analyze non-hermitean matrices, and then introduce the Method of Hermitization, which solves this problem. Then, for rotationally invariant ensembles, I derive a master equation for the average density of eigenvalues in the complex plane, in the limit of infinitely large matrices. This is achieved by resumming all the planar diagrams which appear in the perturbative expansion of the hermitized Green function. Remarkably, this resummation can be carried {\em explicitly} for any rotationally invariant ensemble. I prove that in the limit of infinitely large matrices, the shape of the eigenvalue distribution is either a disk or an annulus. This is the celebrated ``Single-Ring'' Theorem. Which of these shapes is realized is determined by the parameters (coupling constants) which determine the ensemble. By varying these parameters a phase transition may occur between the two possible shapes. I briefly discuss the universal features of this transition. As the analysis of this problem relies heavily on summation of planar Feynman diagrams, I take special effort at presenting a pedagogical exposition of the diagrammatic method, which some readers may find useful.

Quantum Chaos in Supersymmetric QCD at Finite Density

We investigate the distribution of the spacings of adjacent eigenvalues of the lattice Dirac operator. At zero chemical potential $\mu$, the nearest-neighbor spacing distribution $P(s)$ follows the Wigner surmise of random matrix theory both in the confinement and in the deconfinement phase. This is indicative of quantum chaos. At nonzero chemical potential, the eigenvalues of the Dirac operator become complex and we discuss how $P(s)$ can be defined in the complex plane. Numerical results from an SU(2) simulation with staggered fermions in fundamental and adjoint representations are compared with predictions from non-hermitian random matrix theory, and agreement with the Ginibre ensemble is found for $\mu\approx 0.5$.

Eigenvalue Distribution in the Self-Dual Non-Hermitian Ensemble

We consider an ensemble of self-dual matrices with arbitrary complex entries. This ensemble is closely related to a previously defined ensemble of anti-symmetric matrices with arbitrary complex entries. We study the two-level correlation functions numerically. Although no evidence of non-monotonicity is found in the real space correlation function, a definite shoulder is found. On the analytical side, we discuss the relationship between this ensemble and the β=4 two-dimensional one-component plasma, and also argue that this ensemble, combined with other ensembles, exhausts the possible universality classes in non-hermitian random matrix theory. This argument is based on combining the method of hermitization of Feinberg and Zee with Zirnbauer's classification of ensembles in terms of symmetric spaces.

Non-Hermitian Random Matrix Theory and Lattice QCD with Chemical Potential

In quantum chromodynamics (QCD) at nonzero chemical potential, the eigenvalues of the Dirac operator are scattered in the complex plane. Can the fluctuation properties of the Dirac spectrum be described by universal predictions of non-Hermitian random matrix theory? We introduce an unfolding procedure for complex eigenvalues and apply it to data from lattice QCD at finite chemical potential $\mu$ to construct the nearest-neighbor spacing distribution of adjacent eigenvalues in the complex plane. For intermediate values of $\mu$, we find agreement with predictions of the Ginibre ensemble of random matrix theory, both in the confinement and in the deconfinement phase.

Quantum chaos and QCD at finite chemical potential

We investigate the distribution of the spacings of adjacent eigenvalues of the lattice Dirac operator. At zero chemical potential μ, the nearest-neighbor spacing distribution P( s) follows the Wigner surmise of random matrix theory both in the confinement and in the deconfinement phase. This is indicative of quantum chaos. At nonzero chemical potential, the eigenvalues of the Dirac operator become complex. We discuss how P( s) can be defined in the complex plane. Numerical results from an SU(3) simulation with staggered fermions are compared with predictions from non-hermitian random matrix theory, and agreement with the Ginibre ensemble is found for μ ≈ 0.7.

Non-hermitian random matrix theory: Method of hermitian reduction

We consider random non-hermitian matrices in the large- N limit. The power of analytic function theory cannot be brought to bear directly to analyze non-hermitian random matrices, in contrast to hermitian random matrices. To overcome this difficulty, we show that associated to each ensemble of non-hermitian matrices there is an auxiliary ensemble of random hermitian matrices which can be analyzed by the usual methods. We then extract the Green function and the density of eigenvalues of the non-hermitian ensemble from those of the auxiliary ensemble. We apply this “method of hermitization” to several examples, and discuss a number of related issues.

Non-gaussian non-hermitian random matrix theory: Phase transition and addition formalism

We apply the recently introduced method of hermitization to study in the large N limit non-hermitian random matrices that are drawn from a large class of circularly symmetric non-gaussian probability distributions, thus extending the recent gaussian non-hermitian literature. We develop the general formalism for calculating the Green function and averaged density of eigenvalues, which may be thought of as the non-hermitian analog of the method due to Brèzin, ltzykson, Parisi and Zuber for analyzing hermitian non-gaussian random matrices. We obtain an explicit algebraic equation for the integrated density of eigenvalues. A somewhat surprising result of that equation is that the shape of the eigenvalue distribution in the complex plane is either a disk or an annulus. As a concrete example, we analyze the quartic ensemble and study the phase transition from a disk shaped eigenvalue distribution to an annular distribution. Finally, we apply the method of hermitization to develop the addition formalism for free non-hermitian random variables. We use this formalism to state and prove a non-abelian non-hermitian version of the central limit theorem.