Gaussian Random Matrix Ensembles Research Articles

Winding number statistics for chiral random matrices: Averaging ratios of parametric determinants in the orthogonal case

We extend our recent study of winding number density statistics in Gaussian random matrix ensembles of the chiral unitary (AIII) and chiral symplectic (CII) classes. Here, we consider the chiral orthogonal (BDI) case which is the mathematically most demanding one. The key observation is that we can map the topological problem on a spectral one, rendering the toolbox of random matrix theory applicable. In particular, we employ a technique that exploits supersymmetry structures without reformulating the problem in superspace.

Sensitivity of (multi)fractal eigenstates to a perturbation of the Hamiltonian

We study the response of an isolated quantum system governed by the Hamiltonian drawn from the Gaussian Rosenzweig-Porter random matrix ensemble to a perturbation controlled by a small parameter. We focus on the density of states, local density of states and the eigenfunction amplitude overlap correlation functions which are calculated exactly using the mapping to the supersymmetric nonlinear sigma model. We show that the susceptibility of eigenfunction fidelity to the parameter of perturbation can be expressed in terms of these correlation functions and is strongly peaked at the localization transition: It is independent of the effective disorder strength in the ergodic phase, grows exponentially with increasing disorder in the fractal phase and decreases exponentially in the localized phase. As a function of the matrix size, the fidelity susceptibility remains constant in the ergodic phase and increases in the fractal and in the localized phases at modestly strong disorder. We show that there is a critical disorder strength inside the insulating phase such that for disorder stronger than the critical the fidelity susceptibility decreases with increasing the system size. The overall behavior is very similar to the one observed numerically in a recent work by Sels and Polkovnikov [Phys. Rev. E 104, 054105 (2021)] for the normalized fidelity susceptibility in a disordered XXZ spin chain.

Many body density of states in the edge of the spectrum: non-interacting limit

In noninteracting limit, the density of states (dos) of a many body system can be expressed as a convolution of the single body dos of its subunits. We use the formulation to derive, in the edge of the spectrum, a differential equation for the ensemble averaged many body dos that is relatively easier to solve. Our analysis, based on the systems in which the subunits can be modelled by a Gaussian or Wishart random matrix ensemble, indicates that a rescaling of energy by the number of subunits leaves the many body dos in a mathematically invariant form.

Detecting few-body quantum chaos: out-of-time ordered correlators at saturation

We study numerically and analytically the time dependence and saturation of out-of-time ordered correlators (OTOC) in chaotic few-body quantum-mechanical systems: quantum Henon-Heiles system (weakly chaotic), BMN matrix quantum mechanics (strongly chaotic) and Gaussian random matrix ensembles. The growth pattern of quantum-mechanical OTOC is complex and nonuniversal, with no clear exponential regime at relevant timescales in any of the examples studied (which is not in contradiction to the exponential growth found in the literature for many-body systems, i.e. fields). On the other hand, the plateau (saturated) value of OTOC reached at long times decreases with temperature in a simple and universal way: exp(const./T2) for strong chaos (including random matrices) and exp(const./T) for weak chaos. For small matrices and sufficiently complex operators, there is also another, high-temperature regime where the saturated OTOC grows with temperature. Therefore, the plateau OTOC value is a meaningful indicator of few-body quantum chaos. We also discuss some general consequences of our findings for the AdS/CFT duality.

Deviations from Poisson statistics in the spectra of free rectangular thin plates.

The counterintuitive fact that wave chaos appears in the bending spectrum of free rectangular thin plates is presented. After extensive numerical simulations, varying the ratio between the length of its sides, it is shown that (i) frequency levels belonging to different symmetry classes cross each other and (ii) for levels within the same symmetry sector, only avoided crossings appear. The consequence of anticrossings is studied by calculating the distribution of the ratio of consecutive level spacings for each symmetry class. The resulting ratio distributions disagree with the expected Poissonian result. They are then compared with some well-known transition distributions between Poisson and the Gaussian orthogonal random matrix ensemble. It is found that the distribution of the ratio of consecutive level spacings agrees with the prediction of the Rosenzweig-Porter model. Also, the normal-mode vibration amplitudes are found experimentally on aluminum plates, before and after an avoided crossing for symmetrical-symmetrical, symmetrical-antisymmetrical, and antisymmetrical-symmetrical classes. The measured modes show an excellent agreement with our numerical predictions. The expected Poissonian distribution is recovered for the simply supported rectangular plate.

Universal distributions from non-Hermitian perturbation of zero modes.

Hermitian operators with exact zero modes subject to non-Hermitian perturbations are considered. Specific focus is on the distribution of the former zero eigenvalues of the Hermitian operators. The broadening of these zero modes is found to follow an elliptic Gaussian random matrix ensemble of fixed size, where the symmetry class of the perturbation determines the behavior of the modes. This distribution follows from a central limit theorem of matrices and is shown to be robust to deformations.

Universal broadening of zero modes: A general framework and identification.

We consider the smallest eigenvalues of perturbed Hermitian operators with zero modes, either topological or system specific. To leading order for small generic perturbation we show that the corresponding eigenvalues broaden to a Gaussian random matrix ensemble of size ν×ν, where ν is the number of zero modes. This observation unifies and extends a number of results within chiral random matrix theory and effective field theory and clarifies under which conditions they apply. The scaling of the former zero modes with the volume differs from the eigenvalues in the bulk, which we propose as an indicator to identify them in experiments. These results hold for all 10 symmetric spaces in the Altland-Zirnbauer classification and build on two facts. First, the broadened zero modes decouple from the bulk eigenvalues and, second, the mixing from eigenstates of the perturbation form a central limit theorem argument for matrices.

Random supermatrices with an external source

In the past we have considered Gaussian random matrix ensembles in the presence of an external matrix source. The reason was that it allowed, through an appropriate tuning of the eigenvalues of the source, to obtain results on non-trivial dual models, such as Kontsevich’s Airy matrix models and generalizations. The techniques relied on explicit computations of the k-point functions for arbitrary N (the size of the matrices) and on an N-k duality. Numerous results on the intersection numbers of the moduli space of curves were obtained by this technique. In order to generalize these results to include surfaces with boundaries, we have extended these techniques to supermatrices. Again we have obtained quite remarkable explicit expressions for the k-point functions, as well as a duality. Although supermatrix models a priori lead to the same matrix models of 2d-gravity, the external source extensions considered in this article lead to new geometric results.

Robustness of optimal transport in disordered interacting many-body networks.

The robustness of quantum transport under various perturbations is analyzed in disordered interacting many-body systems, which are constructed from the embedded Gaussian random matrix ensembles (EGEs). The transport efficiency can be enhanced drastically, if centrosymmetry (csEGE) is imposed. When the csEGE is perturbed with an ordinary EGE, the transport efficiency in the optimal cases is reduced significantly, while in the suboptimal cases the changes are less pronounced. Qualitatively the same behavior is observed, when parity and centrosymmetry are broken by block perturbations. Analyzing the influence of the environment coupling, optimal transport is observed at a certain coupling strength, while too weak and too strong coupling reduce the transport. Taking into account the effects of decoherence, in the EGE the transport efficiency approaches its maximum at a finite nonzero decoherence strength (environment-assisted transport). In the csEGE the efficiency decays monotonically with the decoherence but is always larger than in the EGE.

Level crossing in random matrices: I. Random perturbation of a fixed matrix

We consider level crossing in a matrix family where H0 is a fixed matrix and V belongs to one of the standard Gaussian random matrix ensembles. We study the probability distribution of level crossing points in the complex plane of , for which we obtain a number of exact, asymptotic and approximate formulas.

Efficient quantum transport in disordered interacting many-body networks.

The coherent transport of n fermions in disordered networks of l single-particle states connected by k-body interactions is studied. These networks are modeled by embedded Gaussian random matrix ensemble (EGE). The conductance bandwidth and the ensemble-averaged total current attain their maximal values if the system is highly filled n∼l-1 and k∼n/2. For the cases k=1 and k=n the bandwidth is minimal. We show that for all parameters the transport is enhanced significantly whenever centrosymmetric embedded Gaussian ensemble (csEGE) are considered. In this case the transmission shows numerous resonances of perfect transport. Analyzing the transmission by spectral decomposition, we find that centrosymmetry induces strong correlations and enhances the extrema of the distributions. This suppresses destructive interference effects in the system and thus causes backscattering-free transmission resonances that enhance the overall transport. The distribution of the total current for the csEGE has a very large dominating peak for n=l-1, close to the highest observed currents.

Local Spectral Statistics of Gaussian Matrices with Correlated Entries.

We prove optimal local law, bulk universality and non-trivial decay for the off-diagonal elements of the resolvent for a class of translation invariant Gaussian random matrix ensembles with correlated entries.

Universality in the Full Counting Statistics of Trapped Fermions

We study the distribution of particle number in extended subsystems of a one-dimensional noninteracting Fermi gas confined in a potential well at zero temperature. Universal features are identified in the scaled bulk and edge regions of the trapped gas where the full counting statistics are given by the corresponding limits of the eigenvalue statistics in Gaussian unitary random matrix ensembles. The universal limiting behavior is confirmed by the bulk and edge scaling of the particle number fluctuations and the entanglement entropy.

Onset of chaos in finite interacting Boson systems with F-spin

For m bosons, carrying spin (f = 1 2) degree of freedom, in Ω single particle levels which are doubly degenerate, one-plus two-body Embedded Gaussian Orthogonal random matrix Ensemble which conserves spin (F) [BEGOE(1 + 2)-F] is defined and spectral properties are examined. Using mean field defined by random single particle energies, for fixed-(m,F) BEGOE(1 + 2)-F, density of states being close to Gaussian, we analyze spectral fluctuation properties, like Nearest Neighbour Spacing Distribution (NNSD) and Dyson-Mehta (Δ3) statistic, as a function of λ- the two-body interaction strength. Numerical calculations are used to demonstrate that for very small value of λ, NNSD and Δ3 are found close to Poisson form and they move steadily to GOE form as λ increases. Moreover, spin dependence of the transition point λ C is obtained. It is also verified using periodogram analysis that BEGOE(1 + 2)-F spectra shows 1/f–noise behaviour.

Supersymmetric virial expansion for time-reversal invariant disordered systems

We develop a supersymmetric virial expansion for two-point correlation functions of almost diagonal Gaussian random matrix ensembles (ADRMT) of the orthogonal symmetry. These ensembles have multiple applications in physics and can be used to study the universal properties of time-reversal invariant disordered systems which are either insulators or close to the Anderson localization transition. We derive a two-level contribution to the correlation functions of the generic ADRMT and apply these results to the critical (multifractal) power law banded ADRMT. Analytical results are compared with numerical ones.

Two-eigenfunction correlation in a multifractal metal and insulator

We consider the correlation of two single-particle probability densities $|\Psi_{E}({\bf r})|^{2}$ at coinciding points ${\bf r}$ as a function of the energy separation $\omega=|E-E'|$ for disordered tight-binding lattice models (the Anderson models) and certain random matrix ensembles. We focus on the models in the parameter range where they are close but not exactly at the Anderson localization transition. We show that even far away from the critical point the eigenfunction correlation show the remnant of multifractality which is characteristic of the critical states. By a combination of the numerical results on the Anderson model and analytical and numerical results for the relevant random matrix theories we were able to identify the Gaussian random matrix ensembles that describe the multifractal features in the metal and insulator phases. In particular those random matrix ensembles describe new phenomena of eigenfunction correlation we discovered from simulations on the Anderson model. These are the eigenfunction mutual avoiding at large energy separations and the logarithmic enhancement of eigenfunction correlations at small energy separations in the two-dimensional (2D) and the three-dimensional (3D) Anderson insulator. For both phenomena a simple and general physical picture is suggested.

Power-law eigenvalue density, scaling, and critical random-matrix ensembles

We consider a class of rotationally invariant unitary random matrix ensembles where the eigenvalue density falls off as an inverse power law. Under a scaling appropriate for such power-law densities (different from the scaling required in Gaussian random matrix ensembles), we calculate exactly the two-level kernel that determines all eigenvalue correlations. We show that such ensembles belong to the class of critical ensembles.

Level repulsion in constrained Gaussian random-matrix ensembles

Introducing sets of constraints, we define new classes of random-matrix ensembles, the constrained Gaussian unitary (CGUE) and the deformed Gaussian unitary (DGUE) ensembles. The latter interpolate between the GUE and the CGUE. We derive a sufficient condition for GUE-type level repulsion to persist in the presence of constraints. For special classes of constraints, we extend this approach to the orthogonal and to the symplectic ensembles. A generalized Fourier theorem relates the spectral properties of the constraining ensembles with those of the constrained ones. We find that in the DGUEs, level repulsion always prevails at a sufficiently short distance and may be lifted only in the limit of strictly enforced constraints.

Asymptotic form of the density profile for Gaussian and Laguerre random matrix ensembles with orthogonal and symplectic symmetry

In a recent study we have obtained correction terms to the large N asymptotic expansions of the eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of random N×N matrices, both in the bulk and at the soft edge of the spectrum. In the present study these results are used to similarly analyze the eigenvalue density for Gaussian and Laguerre random matrix ensembles with orthogonal and symplectic symmetry. As in the case of unitary symmetry, a matching is exhibited between the asymptotic expansion of the bulk density, expanded about the edge, and the asymptotic expansion of the edge density, expanded into the bulk. In addition, aspects of the asymptotic expansion of the smoothed density, which involves delta functions at the endpoints of the support, are interpreted microscopically.

On the Gaussian Random Matrix Ensembles with Additional Symmetry Conditions

The Gaussian unitary random matrix ensembles satisfying some additional sym- metry conditions are considered. The eect of these conditions on the limiting normalized counting measures and correlation functions is studied.