arXiv subjects
Publications and source records attributed to Yan V. Fyodorov.
This paper is a detailed account of the recent progress in understanding the statistical properties of complex eigenvalues of random non-Hermitian matrices reported earlier in our two short communications: Physics Letters A v.226, 46 (1997) and Phys. Rev. Lett., v.79, 557 (1997)
Using the random matrix description of open quantum chaotic systems we calculate in closed form the universal autocorrelation function and the probability distribution of the total photodissociation cross section in the regime of quantum chaos.
It is generally accepted that statistics of energy levels in closed chaotic quantum systems is adequately described by the theory of Random Hermitian Matrices. Much less is known about properties of "resonances" - generic features of open quantum systems pertinent for understanding of scattering experiments. In the framework of the Heidelberg approach to quantum chaotic scattering open systems are characterized by an effective non-Hermitian random matrix Hamiltonian $\hat{H}_{ef}$. Complex eigenvalues of $\hat{H}_{ef}$ are S-matrix poles (resonances). We show how to find the mean density of these poles (Fyodorov and Sommers) and how to use the effective Hamiltonian to calculate autocorrelations of the photodissociation cross section (Fyodorov and Alhassid). In the second part of the paper we review recent results (Fyodorov, Khoruzhenko and Sommers) on non-Hermitian matrices with independent entries in the regime of weak Non-Hermiticity. This regime describes a crossover from Hermitian matrices characterized by Wigner-Dyson statistics of real eigenvalues to strongly non-Hermitian ones whose complex eigenvalues were studied by Ginibre.
We study fluctuations of the local density of states (LDOS) on a tree-like lattice with large branching number $m$. The average form of the local spectral function (at given value of the random potential in the observation point) shows a crossover from the Lorentzian to semicircular form at $α\sim 1/m$, where $α= (V/W)^2$, $V$ is the typical value of the hopping matrix element, and $W$ is the width of the distribution of random site energies. For $α>1/m^2$ the LDOS fluctuations (with respect to this average form) are weak. In the opposite case, $α<1/m^2$, the fluctuations get strong and the average LDOS ceases to be representative, which is related to the existence of the Anderson transition at $α_c\sim 1/(m^2\log^2m)$. On the localized side of the transition the spectrum is discrete, and LDOS is given by a set of $δ$-like peaks. The effective number of components in this regime is given by $1/P$, with $P$ being the inverse participation ratio. It is shown that $P$ has in the transition point a limiting value $P_c$ close to unity, $1-P_c\sim 1/\log m$, so that the system undergoes a transition directly from the deeply localized to extended phase. On the side of delocalized states, the peaks in LDOS get broadened, with a width $\sim\exp\{-{const}\log m[(α-α_c)/α_c]^{-1/2}\}$ being exponentially small near the transition point. We discuss application of our results to the problem of the quasiparticle line shape in a finite Fermi system, as suggested recently by Altshuler, Gefen, Kamenev, and Levitov.
By using the method of orthogonal polynomials we analyze the statistical properties of complex eigenvalues of random matrices describing a crossover from Hermitian matrices characterized by the Wigner- Dyson statistics of real eigenvalues to strongly non-Hermitian ones whose complex eigenvalues were studied by Ginibre. Two-point statistical measures (as e.g. spectral form factor, number variance and small distance behavior of the nearest neighbor distance distribution $p(s)$) are studied in more detail. In particular, we found that the latter function may exhibit unusual behavior $p(s)\propto s^{5/2}$ for some parameter values.
We analyse universal statistical properties of phase shifts and time delays for open chaotic systems in the crossover regime of partly broken time-reversal invariance. In particular, we find that the distribution of the time delay shows $τ^{-3/2}$ behavior for weakly open systems of any symmetry.
We study overlap of two different eigenfunctions as compared with self-overlap in the framework of an infinite-dimensional version of the disordered tight-binding model. Despite a very sparse structure of the eigenstates in the vicinity of Anderson transition their mutual overlap is still found to be of the same order as self-overlap as long as energy separation is smaller than a critical value. The latter fact explains robustness of the Wigner-Dyson level statistics everywhere in the phase of extended states. The same picture is expected to hold for usual d-dimensional conductors, ensuring the $s^β$ form of the level repulsion at critical point.
We study numerically and analytically the moments of the dimensionless level curvature for one-dimensional disordered rings of the circumference L pierced by a magnetic flux. The negative moments of the curvature distribution can be evaluated analytically in the extreme localization limit. The ensuing small curvature asymptotic of the corresponding distribution has a "log-normal" behavior. Numerically studied positive moments show differences from other log-normally distributed quantities.
We consider an ensemble of large non-Hermitian random matrices of the form $\hat{H}+i\hat{A}_s$, where $\hat{H}$ and $\hat{A}_s$ are Hermitian statistically independent random $N\times N$ matrices. We demonstrate the existence of a new nontrivial regime of weak non-Hermiticity characterized by the condition that the average of $N\mbox{Tr} \hat{A}_s^2$ is of the same order as that of $\mbox{Tr} \hat{H}^2 $ when $N\to \infty$. We find explicitly the density of complex eigenvalues for this regime in the limit of infinite matrix dimension. The density determines the eigenvalue distribution in the crossover regime between random Hermitian matrices whose real eigenvalues are distributed according to the Wigner semi-circle law and random complex matrices whose eigenvalues are distributed in the complex plane according to the so-called ``elliptic law''.
We study statistical properties of the ensemble of large $N\times N$ random matrices whose entries $ H_{ij}$ decrease in a power-law fashion $H_{ij}\sim|i-j|^{-α}$. Mapping the problem onto a nonlinear $σ-$model with non-local interaction, we find a transition from localized to extended states at $α=1$. At this critical value of $α$ the system exhibits multifractality and spectral statistics intermediate between the Wigner-Dyson and Poisson one. These features are reminiscent of those typical for the mobility edge of disordered conductors. We find a continuous set of critical theories at $α=1$, parametrized by the value of the coupling constant of the $σ-$model. At $α>1$ all states are expected to be localized with integrable power-law tails. At the same time, for $1<α<3/2$ the wave packet spreading at short time scale is superdiffusive: $\langle |r|\rangle\sim t^{\frac{1}{2α-1}}$, which leads to a modification of the Altshuler-Shklovskii behavior of the spectral correlation function. At $1/2<α<1$ the statistical properties of eigenstates are similar to those in a metallic sample in $d=(α-1/2)^{-1}$ dimensions. Finally, the region $α<1/2$ is equivalent to the corresponding Gaussian ensemble of random matrices $(α=0)$. The theoretical predictions are compared with results of numerical simulations.
By using the supersymmetry method we derive an explicit expression for the parametric correlation function of densities of eigenphases $θ_a$ of the S-matrix in a chaotic quantum system with broken time-reversal symmetry coupled to continua via M equivalent open channels;$\,\,a=1,..,M$ .We use it to find the distribution of derivatives of these eigenphases over the energy ("phaseshift times") as well as over an arbitrary external parameter.We also find the parametric correlations of Wigner-Smith delay times
We derive the explicit expression for the distribution of resonance widths in a chaotic quantum system coupled to continua via M equivalent open channels. It describes a crossover from the $χ^2$ distribution (regime of isolated resonances) to a broad power-like distribution typical for the regime of overlapping resonances. The first moment is found to reproduce exactly the Moldauer-Simonius relation between the mean resonance width and the transmission coefficient. This fact may serve as another manifestation of equivalence between the spectral and the ensemble averaging.
The random banded matrices (RBM) whose diagonal elements fluctuate much stronger than the off-diagonal ones were introduced recently by Shepelyansky as a convenient model for coherent propagation of two interacting particles in a random potential. We treat the problem analytically by using the mapping onto the same supersymmetric nonlinear $σ-$model that appeared earlier in consideration of the standard RBM ensemble, but with renormalized parameters. A Lorentzian form of the local density of states and a two-scale spatial structure of the eigenfunctions revealed recently by Jacquod and Shepelyansky are confirmed by direct calculation of the distribution of eigenfunction components.
We calculate analytically the distributions of "level curvatures" (LC) (the second derivatives of eigenvalues with respect to a magnetic flux) for a particle moving in a white-noise random potential. We find that the Zakrzewski-Delande conjecture is still valid even if the lowest weak localization corrections are taken into account. The ratio of mean level curvature modulus to mean dissipative conductance is proved to be universal and equal to $2π$ in agreement with available numerical data.
We calculate the distribution of eigenfunction amplitudes and the variance of the ``inverse participation ratio'' (IPR) in disordered metallic samples. A relation is established between the distribution function of IPR and that of ``level velocities'' -- parametric derivatives of energy levels with respect to a random perturbation.
The explicit analytical expression for the distribution function of parametric derivatives of energy levels ("level velocities") with respect to a random change of scattering potential is derived for the chaotic quantum systems belonging to the quasi 1D universality class (quantum kicked rotator, "domino" billiard, disordered wire, etc.).