arXiv subjects

Mario Kieburg

Publications and source records attributed to Mario Kieburg.

At least 37 records · Page 2Linked to original sources

arXiv2023

A rate of convergence when generating stable invariant Hermitian random matrix ensembles

Recently, we have classified Hermitian random matrix ensembles that are invariant under the conjugate action of the unitary group and stable with respect to matrix addition. Apart from a scaling and a shift, the whole information of such an ensemble is encoded in the stability exponent determining the ``heaviness'' of the tail and the spectral measure that describes the anisotropy of the probability distribution. In the present work, we address the question how these ensembles can be generated by the knowledge of the latter two quantities. We consider a sum of a specific construction of identically and independently distributed random matrices that are based on Haar distributed unitary matrices and a stable random vectors. For this construction, we derive the rate of convergence in the supremums norm and show that this rate is optimal in the class of all stable invariant random matrices for a fixed stability exponent. As a consequence we also give the rate of convergence in the total variation distance.

math.PR↗

arXiv2023

Winding Number Statistics for Chiral Random Matrices: Averaging Ratios of Determinants with Parametric Dependence

Topological invariance is a powerful concept in different branches of physics as they are particularly robust under perturbations. We generalize the ideas of computing the statistics of winding numbers for a specific parametric model of the chiral Gaussian Unitary Ensemble to other chiral random matrix ensembles. Especially, we address the two chiral symmetry classes, unitary (AIII) and symplectic (CII), and we analytically compute ensemble averages for ratios of determinants with parametric dependence. To this end, we employ a technique that exhibits reminiscent supersymmetric structures while we never carry out any map to superspace.

math-ph↗

arXiv2022

Derivative principles for invariant ensembles

In the present work we show that the joint probability distribution of the eigenvalues can be expressed in terms of a differential operator acting on the distribution of some other matrix quantities. Those quantities might be the diagonal or pseudo-diagonal entries as it is the case for Hermitian matrices. These representations are called derivative principles. We show them for the additive spaces of the Hermitian, Hermitian antisymmetric, Hermitian anti-self-dual, and complex rectangular matrices as well as for the two multiplicative matrix spaces of the positive definite Hermitian matrices and of the unitary matrices. In all six cases we prove the uniqueness of the derivative principles.

math-ph↗

arXiv2022

Volume-law entanglement entropy of typical pure quantum states

The entanglement entropy of subsystems of typical eigenstates of quantum many-body Hamiltonians has been recently conjectured to be a diagnostic of quantum chaos and integrability. In quantum chaotic systems it has been found to behave as in typical pure states, while in integrable systems it has been found to behave as in typical pure Gaussian states. In this tutorial, we provide a pedagogical introduction to known results about the entanglement entropy of subsystems of typical pure states and of typical pure Gaussian states. They both exhibit a leading term that scales with the volume of the subsystem, when smaller than one half of the volume of the system, but the prefactor of the volume law is fundamentally different. It is constant (and maximal) for typical pure states, and it depends on the ratio between the volume of the subsystem and of the entire system for typical pure Gaussian states. Since particle-number conservation plays an important role in many physical Hamiltonians, we discuss its effect on typical pure states and on typical pure Gaussian states. We prove that while the behavior of the leading volume-law terms does not change qualitatively, the nature of the subleading terms can change. In particular, subleading corrections can appear that depend on the square root of the volume of the subsystem. We unveil the origin of those corrections. Finally, we discuss the connection between the entanglement entropy of typical pure states and analytical results obtained in the context of random matrix theory, as well as numerical results obtained for physical Hamiltonians.

quant-ph↗

arXiv2022

Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics

The framework of spherical transforms and Pólya ensembles is of utility in deriving structured analytic results for sums and products of random matrices in a unified way. In the present work, we will carry over this framework to study products of unitary matrices. Those are not distributed via the Haar measure, but still are drawn from distributions where the eigenvalue and eigenvector statistics factorise. They include the circular Jacobi ensemble, known in relation to the Fisher-Hartwig singularity in the theory of Toeplitz determinants, as well as the heat kernel for Brownian motion on the unitary group. We define cyclic Pólya frequency functions and show their relation to the cyclic Pólya ensembles, give a uniqueness statement for the corresponding weights, and derive the determinantal point processes of the eigenvalue statistics at fixed matrix dimension. An outline is given of problems one may encounter when investigating the local spectral statistics.

math.PR↗

arXiv2022

Hard Edge Statistics of Products of Pólya Ensembles and Shifted GUE's

Very recently, we have shown how the harmonic analysis approach can be modified to deal with products of general Hermitian and complex random matrices at finite matrix dimension. In the present work, we consider the particular product of a multiplicative Pólya ensemble on the complex square matrices and a Gaussian unitary ensemble (GUE) shifted by a constant multiplicative of the identity. The shift shall show that the limiting hard edge statistics of the product matrix is weakly dependent on the local spectral statistics of the GUE, but depends more on the global statistics via its Stieltjes transform (Green function). Under rather mild conditions for the Pólya ensemble, we prove formulas for the hard edge kernel of the singular value statistics of the Pólya ensemble alone and the product matrix to highlight their very close similarity. Due to these observations, we even propose a conjecture for the hard edge statistics of a multiplicative Pólya ensemble on the complex matrices and a polynomial ensemble on the Hermitian matrices.

math.PR↗

arXiv2021

Harmonic analysis for rank-1 Randomised Horn Problems

The randomised Horn problem, in both its additive and multiplicative version, has recently drawn increasing interest. Especially, closed analytical results have been found for the rank-1 perturbation of sums of Hermitian matrices and products of unitary matrices. We will generalise these results to rank-1 perturbations for products of positive definite Hermitian matrices and prove the other results in a new unified way. Our ideas work along harmonic analysis for matrix groups via spherical transforms that have been successfully applied in products of random matrices in the past years. In order to achieve the unified derivation of all three cases, we define the spherical transform on the unitary group and prove its invertibility.

math-ph↗

arXiv2021

Cascade of phase transitions in a planar Dirac material

We investigate a model of interacting Dirac fermions in $2+1$ dimensions with $M$ flavors and $N$ colors having the $\mathrm{U}(M)\times \mathrm{SU}(N)$ symmetry. In the large-$N$ limit, we find that the $\mathrm{U}(M)$ symmetry is spontaneously broken in a variety of ways. In the vacuum, when the parity-breaking flavor-singlet mass is varied, the ground state undergoes a sequence of $M$ first-order phase transitions, experiencing $M+1$ phases characterized by symmetry breaking $\mathrm{U}(M)\to \mathrm{U}(M-k)\times \mathrm{U}(k)$ with $k\in\{0,1,2,\cdots,M\}$, bearing a close resemblance to the vacuum structure of three-dimensional QCD. At finite temperature and chemical potential, a rich phase diagram with first and second-order phase transitions and tricritical points is observed. Also exotic phases with spontaneous symmetry breaking of the form as $\mathrm{U}(3)\to \mathrm{U}(1)^3$, $\mathrm{U}(4)\to \mathrm{U}(2)\times \mathrm{U}(1)^2$, and $\mathrm{U}(5)\to \mathrm{U}(2)^2\times \mathrm{U}(1)$ exist. For a large flavor-singlet mass, the increase of the chemical potential $μ$ brings about $M$ consecutive first-order transitions that separate the low-$μ$ phase diagram with vanishing fermion density from the high-$μ$ region with a high fermion density.

hep-th↗

arXiv2021

The Page Curve for Fermionic Gaussian States

In a seminal paper, Page found the exact formula for the average entanglement entropy for a pure random state. We consider the analogous problem for the ensemble of pure fermionic Gaussian states, which plays a crucial role in the context of random free Hamiltonians. Using recent results from random matrix theory, we show that the average entanglement entropy of pure random fermionic Gaussian states in a subsystem of $N_A$ out of $N$ degrees of freedom is given by $\langle S_A\rangle_\mathrm{G}\!=\!(N\!-\!\tfrac{1}{2})Ψ(2N)\!+\!(\tfrac{1}{4}\!-\!N_A)Ψ(N)\!+\!(\tfrac{1}{2}\!+\!N_A\!-\!N)Ψ(2N\!-\!2N_A)\!-\!\tfrac{1}{4}Ψ(N\!-\!N_A)\!-\!N_A$, where $Ψ$ is the digamma function. Its asymptotic behavior in the thermodynamic limit is given by $\langle S_A\rangle_\mathrm{G}\!=\! N(\log 2-1)f+N(f-1)\log(1-f)+\tfrac{1}{2}f+\tfrac{1}{4}\log{(1-f)}\,+\,O(1/N)$, where $f=N_A/N$. Remarkably, its leading order agrees with the average over eigenstates of random quadratic Hamiltonians with number conservation, as found by Lydzba, Rigol and Vidmar. Finally, we compute the variance in the thermodynamic limit, given by the constant $\lim_{N\to\infty}(ΔS_A)^2_{\mathrm{G}}=\frac{1}{2}(f+f^2+\log(1-f))$.

quant-ph↗

arXiv2021

Local Tail Statistics of Heavy-Tailed Random Matrix Ensembles with Unitary Invariance

We study heavy-tailed Hermitian random matrices that are unitarily invariant. The invariance implies that the eigenvalue and eigenvector statistics are decoupled. The motivating question has been whether a freely stable random matrix has stable eigenvalue statistics for the largest eigenvalues in the tail. We investigate this question through the use of both numerical and analytical means, the latter of which makes use of the supersymmetry method. A surprising behaviour is uncovered in that a freely stable random matrix does not necessarily yield stable statistics and if it does then it might exhibit Poisson or Poisson-like statistics. The Poisson statistics have been already observed for heavy-tailed Wigner matrices. We conclude with two conjectures on this peculiar behaviour.

math-ph↗

arXiv2020

Universality of local spectral statistics of products of random matrices

We derive exact analytical expressions for correlation functions of singular values of the product of $M$ Ginibre matrices of size $N$ in the double scaling limit $M,N\rightarrow \infty$. The singular value statistics is described by a determinantal point process with a kernel that interpolates between GUE statistic and Dirac-delta (picket-fence) statistic. In the thermodynamic limit, $N\rightarrow \infty$, the interpolation parameter is given by the limiting quotient $a=N/M$. One of our goals is to find an explicit form of the kernel at the hard edge, in the bulk and at the soft edge for any $a$. We find that in addition to the standard scaling regimes, there is a new transitional regime which interpolates between the hard edge and the bulk. We conjecture that these results are universal, and that they apply to a broad class of products of random matrices from the Gaussian basin of attraction, including correlated matrices. We corroborate this conjecture by numerical simulations. Additionally, we show that the local spectral statistics of the considered random matrix products is identical with the local statistics of Dyson Brownian motion with the initial condition given by equidistant {positions,} with the crucial difference that this equivalence holds only locally. Finally, we have identified a mesoscopic spectral scale at the soft edge which is crucial for the unfolding of the spectrum.

math-ph↗

arXiv2020

Polynomial Ensembles and Pólya Frequency Functions

We study several kinds of polynomial ensembles of derivative type which we propose to call Pólya ensembles. These ensembles are defined on the spaces of complex square, complex rectangular, Hermitian, Hermitian anti-symmetric and Hermitian anti-self-dual matrices, and they have nice closure properties under the multiplicative convolution for the first class and under the additive convolution for the other classes. The cases of complex square matrices and Hermitian matrices were already studied in former works. One of our goals is to unify and generalize the ideas to the other classes of matrices. Here we consider convolutions within the same class of Pólya ensembles as well as convolutions with the more general class of polynomial ensembles. Moreover, we derive some general identities for group integrals similar to the Harish-Chandra-Itzykson-Zuber integral, and we relate Pólya ensembles to Pólya frequency functions. For illustration we give a number of explicit examples for our results.

math.PR↗

arXiv2020

Universal eigenvector correlations in quaternionic Ginibre ensembles

Non-Hermitian random matrices enjoy non-trivial correlations in the statistics of their eigenvectors. We study the overlap among left and right eigenvectors in Ginibre ensembles with quaternion valued Gaussian matrix elements. This concept was introduced by Chalker and Mehlig in the complex Ginibre ensemble. Using a Schur decomposition, for harmonic potentials we can express the overlap in terms of complex eigenvalues only, coming in conjugate pairs in this symmetry class. Its expectation value leads to a Pfaffian determinant, for which we explicitly compute the matrix elements for the induced Ginibre ensemble with $2α$ zero eigenvalues, for finite matrix size $N$. In the macroscopic large-$N$ limit in the bulk of the spectrum we recover the limiting expressions of the complex Ginibre ensemble for the diagonal and off-diagonal overlap, which are thus universal.

math-ph↗

arXiv2019

Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems

We study the transition between integrable and chaotic behaviour in dissipative open quantum systems, exemplified by a boundary driven quantum spin-chain. The repulsion between the complex eigenvalues of the corresponding Liouville operator in radial distance $s$ is used as a universal measure. The corresponding level spacing distribution is well fitted by that of a static two-dimensional Coulomb gas with harmonic potential at inverse temperature $β\in[0,2]$. Here, $β=0$ yields the two-dimensional Poisson distribution, matching the integrable limit of the system, and $β=2$ equals the distribution obtained from the complex Ginibre ensemble, describing the fully chaotic limit. Our findings generalise the results of Grobe, Haake and Sommers who derived a universal cubic level repulsion for small spacings $s$. We collect mathematical evidence for the universality of the full level spacing distribution in the fully chaotic limit at $β=2$. It holds for all three Ginibre ensembles of random matrices with independent real, complex or quaternion matrix elements.

cond-mat.stat-mech↗

arXiv2019

Families of two-dimensional Coulomb gases on an ellipse: correlation functions and universality

We investigate a one-parameter family of Coulomb gases in two dimensions, which are confined to an ellipse, due to a hard wall constraint, and are subject to an additional external potential. At inverse temperature $β=2$ we can use the technique of planar orthogonal polynomials, borrowed from random matrix theory, to explicitly determine all $k$-point correlation functions for a fixed number of particles $N$. These are given by the determinant of the kernel of the corresponding orthogonal polynomials, which in our case are the Gegenbauer polynomials, or a subset of the asymmetric Jacobi polynomials, depending on the choice of external potential, as shown in a companion paper recently published by three of the authors. In the rotationally invariant case, when the ellipse becomes the unit disc, our findings agree with that of the ensemble of truncated unitary random matrices. The thermodynamical large-$N$ limit is investigated in the local scaling regime in the bulk and at the edge of the spectrum at weak and strong non-Hermiticity. We find new universality classes in these limits and recover the sine- and Bessel-kernel in the Hermitian limit. The limiting global correlation functions of particles in the interior of the ellipse are more difficult to obtain but found in the special cases corresponding to the Chebyshev polynomials.

math-ph↗

arXiv2019

Products of Complex Rectangular and Hermitian Random Matrices

Products and sums of random matrices have seen a rapid development in the past decade due to various analytical techniques available. Two of these are the harmonic analysis approach and the concept of polynomial ensembles. Very recently, it has been shown for products of real matrices with anti-symmetric matrices of even dimension that the traditional harmonic analysis on matrix groups developed by Harish-Chandra et al. needs to be modified when considering the group action on general symmetric spaces of matrices. In the present work, we consider the product of complex random matrices with Hermitian matrices, in particular the former can be also rectangular while the latter has not to be positive definite and is considered as a fixed matrix as well as a random matrix. This generalises an approach for products involving the Gaussian unitary ensemble (GUE) and circumvents the use there of non-compact group integrals. We derive the joint probability density function of the real eigenvalues and, additionally, prove transformation formulas for the bi-orthogonal functions and kernels.

math.PR↗

arXiv2019

From Integrable to Chaotic Systems: Universal Local Statistics of Lyapunov exponents

Systems where time evolution follows a multiplicative process are ubiquitous in physics. We study a toy model for such systems where each time step is given by multiplication with an independent random $N\times N$ matrix with complex Gaussian elements, the complex Ginibre ensemble. This model allows to explicitly compute the Lyapunov exponents and local correlations amongst them, when the number of factors $M$ becomes large. While the smallest eigenvalues always remain deterministic, which is also the case for many chaotic quantum systems, we identify a critical double scaling limit $N\sim M$ for the rest of the spectrum. It interpolates between the known deterministic behaviour of the Lyapunov exponents for $M\gg N$ (or $N$ fixed) and universal random matrix statistics for $M\ll N$ (or $M$ fixed), characterising chaotic behaviour. After unfolding this agrees with Dyson's Brownian Motion starting from equidistant positions in the bulk and at the soft edge of the spectrum. This universality statement is further corroborated by numerical experiments, multiplying different kinds of random matrices. It leads us to conjecture a much wider applicability in complex systems, that display a transition from deterministic to chaotic behaviour.

math-ph↗

arXiv2019

Eigenvalue and Eigenvector Statistics in Time Series Analysis

The study of correlated time-series is ubiquitous in statistical analysis, and the matrix decomposition of the cross-correlations between time series is a universal tool to extract the principal patterns of behavior in a wide range of complex systems. Despite this fact, no general result is known for the statistics of eigenvectors of the cross-correlations of correlated time-series. Here we use supersymmetric theory to provide novel analytical results that will serve as a benchmark for the study of correlated signals for a vast community of researchers.

cond-mat.stat-mech↗