Lyapunov Spectra for All Ten Symmetry Classes of Quasi-one-dimensional Disordered Systems of Non-interacting Fermions

A random phase property is proposed for products of random matrices drawn from any one of the classical groups associated with the ten Cartan symmetry classes of non-interacting disordered Fermion systems. It allows to calculate the Lyapunov spectrum explicitly in a perturbative regime. These results apply to quasi-one-dimensional random Dirac operators which can be constructed as representatives for each of the ten symmetry classes. For those symmetry classes that correspond to two-dimensional topological insulators or superconductors, the random Dirac operators describing the one-dimensional boundaries have vanishing Lyapunov exponents and almost surely an absolutely continuous spectrum, reflecting the gapless and conducting nature of the boundary degrees of freedom.     

Effective Field Theories for Disordered Systems from a Generalized Phase Formulation

We consider a spinless particle moving in a d-dimensional box, subject to periodic boundary conditions, and in the presence of a random potential. Introducing the logarithm of the wave function transforms the time-independent Schrödinger equation into a stochastic differential equation with the random potential acting as the source. Using this as our starting point we write functional integral representations for the disorder averaged density of states, the two point correlator of the absolute value of the wave function, and inverse participation ratios. We also show how a deterministic or random magnetic field can be included in the formalism.     

Relaxation to Equilibrium for Two Dimensional Disordered Ising Systems in the Griffiths Phase

We consider Glauber–type dynamics for two dimensional disordered magnets of Ising type. We prove that, if the disorder–averaged influence of the boundary condition is sufficiently small in the equilibrium system, then the corresponding Glauber dynamics is ergodic with probability one and the disorder–average C(t) of time–autocorrelation function satisfies (for large t). For the standard two dimensional dilute Ising ferromagnet with i.i.d. random nearest neighbor couplings taking the values 0 or J ₀>0, our results apply even if the active bonds percolate and J ₀ is larger than the critical value J _c of the corresponding pure Ising model. For the same model we also prove that in the whole Griffiths' phase the previous upper bound is optimal. This implies the existence of a dynamical phase transition which occurs when J crosses J _c . Received:     

Construction of the Lyapunov Spectrum in a Chaotic System Displaying Phase Synchronization

We consider a three-dimensional chaotic system consisting of the suspension of Arnold’s cat map coupled with a clock via a weak dissipative interaction. We show that the coupled system displays a synchronization phenomenon, in the sense that the relative phase between the suspension flow and the clock locks to a special value, thus making the motion fall onto a lower dimensional attractor. More specifically, we construct the attractive invariant manifold, of dimension smaller than three, using a convergent perturbative expansion. Moreover, we compute via convergent series the Lyapunov exponents, including notably the central one. The result generalizes a previous construction of the attractive invariant manifold in a similar but simpler model. The main novelty of the current construction relies in the computation of the Lyapunov spectrum, which consists of non-trivial analytic exponents. Some conjectures about a possible smoothening transition of the attractor as the coupling is increased are also discussed.     

The Lyapunov Spectrum Is Not Always Concave

We characterize one-dimensional compact repellers having non-concave Lyapunov spectra. For linear maps with two branches we give an explicit condition that characterizes non-concave Lyapunov spectra. The first author was partially supported by Proyecto Fondecyt 11070050. Both authors were partially supported by Research Network on Low Dimensional Systems, PBCT/CONICYT, Chile.     

Stretched Exponential Relaxation in Disordered Complex Systems： Fractal Time Random Walk Model

We have analytically derived the relaxation function for one-dimensional disordered complex systems in terms of autocorrelation function of fractal time random walk by using operator formalism. We have shown that the relaxation function has stretched exponential, i.e. the Kohlrausch-Williams-Watts character for a fractal time random walk process.     

Universal Vibrational Properties of Disordered Systems in Terms of the Theory of Random Correlated Matrices

JETP Letters - It has been shown that a correlated Wishart ensemble can be used to study the general vibrational properties of stable amorphous solids, where the energy is translationally...     

Random Matrices and Lyapunov Coefficients Regularity

Analyticity and other properties of the largest or smallest Lyapunov exponent of a product of real matrices with a “cone property” are studied as functions of the matrices entries, as long as they vary without destroying the cone property. The result is applied to stability directions, Lyapunov coefficients and Lyapunov exponents of a class of products of random matrices and to dynamical systems. The results are not new and the method is the main point of this work: it is is based on the classical theory of the Mayer series in Statistical Mechanics of rarefied gases.     

Spectral Spacing Correlations for Chaotic and Disordered Systems

New aspects of spectral fluctuations of (quantum) chaotic and diffusive systems are considered, namely autocorrelations of the spacing between consecutive levels or spacing autocovariances. They can be viewed as a discretized two point correlation function. Their behavior results from two different contributions. One corresponds to (universal) random matrix eigenvalue fluctuations, the other to diffusive or chaotic characteristics of the corresponding classical motion. A closed formula expressing spacing autocovariances in terms of classical dynamical zeta functions, including the Perron–Frobenius operator, is derived. It leads to a simple interpretation in terms of classical resonances. The theory is applied to zeros of the Riemann zeta function. A striking correspondence between the associated classical dynamical zeta functions and the Riemann zeta itself is found. This induces a resurgence phenomenon where the lowest Riemann zeros appear replicated an infinite number of times as resonances and sub-resonances in the spacing autocovariances. The theoretical results are confirmed by existing data. The present work further extends the already well known semiclassical interpretation of properties of Riemann zeros.     

Finite-Time Lyapunov Exponents for Products of Random Transformations

I show how continuous products of random transformations constrained by a generic group structure can be studied by using Iwasawa's decomposition into angular, diagonal, and shear degrees of freedom. In the case of a Gaussian process a set of variables, adapted to the Iwasawa decomposition and still having a Gaussian distribution, is introduced and used to compute the statistics of the finite-time Lyapunov spectrum of the process. The variables also allow to show the exponential freezing of the shear degrees of freedom, which contain information about the Lyapunov eigenvectors.     

Lyapunov Exponents of Random Walks in Small Random Potential: The Lower Bound

We consider the simple random walk on ${\mathbb{Z}^d}$ Z d , d > 3, evolving in a potential of the form β V, where ${(V(x))_{x \in \mathbb{Z}^d}}$ ( V ( x ) ) x ∈ Z d are i.i.d. random variables taking values in [0, + ∞), and β > 0. When the potential is integrable, the asymptotic behaviours as β tends to 0 of the associated quenched and annealed Lyapunov exponents are known (and coincide). Here, we do not assume such integrability, and prove a sharp lower bound on the annealed Lyapunov exponent for small β. The result can be rephrased in terms of the decay of the averaged Green function of the Anderson Hamiltonian ${-\triangle + \beta V}$ - ? + β V .     

Lyapunov Exponents for Products of Complex Gaussian Random Matrices

The exact value of the Lyapunov exponents for the random matrix product P _N =A _N A _N?1?A ₁ with each $A_{i} = \varSigma^{1/2} G_{i}^{\mathrm{c}}$ , where Σ is a fixed d×d positive definite matrix and $G_{i}^{\mathrm{c}}$ a d×d complex Gaussian matrix with entries standard complex normals, are calculated. Also obtained is an exact expression for the sum of the Lyapunov exponents in both the complex and real cases, and the Lyapunov exponents for diffusing complex matrices.     

The Spectrum of Heavy Tailed Random Matrices

Let X _N be an N → N random symmetric matrix with independent equidistributed entries. If the law P of the entries has a finite second moment, it was shown by Wigner [14] that the empirical distribution of the eigenvalues of X _N , once renormalized by , converges almost surely and in expectation to the so-called semicircular distribution as N goes to infinity. In this paper we study the same question when P is in the domain of attraction of an α-stable law. We prove that if we renormalize the eigenvalues by a constant a _N of order , the corresponding spectral distribution converges in expectation towards a law which only depends on α. We characterize and study some of its properties; it is a heavy-tailed probability measure which is absolutely continuous with respect to Lebesgue measure except possibly on a compact set of capacity zero. This work was partially supported by Miller institute for Basic Research in Science, University of California Berkeley.     

Quantum Quenches with Random Matrix Hamiltonians and Disordered Potentials

We numerically investigate statistical ensembles for the occupations of eigenstates of an isolated quantum system emerging as a result of quantum quenches. The systems investigated are sparse random matrix Hamiltonians and disordered lattices. In the former case, the quench consists of sudden switching‐on the off‐diagonal elements of the Hamiltonian. In the latter case, it is sudden switching‐on of the hopping between adjacent lattice sites. The quench‐induced ensembles are compared with the so‐called “quantum micro‐canonical” (QMC) ensemble describing quantum superpositions with fixed energy expectation values. Our main finding is that quantum quenches with sparse random matrices having one special diagonal element lead to the condensation phenomenon predicted for the QMC ensemble. Away from the QMC condensation regime, the overall agreement with the QMC predictions is only qualitative for both random matrices and disordered lattices but with some cases of a very good quantitative agreement. In the case of disordered lattices, the QMC ensemble can be used to estimate the probability of finding a particle in a localized or delocalized eigenstate.     

The Bernoulli Property for Weakly Hyperbolic Systems

A dynamical system is called partially hyperbolic if it exhibits three invariant directions, one unstable (expanding), one stable (contracting) and one central direction (somewhere in between the other two). We prove that topologically mixing partially hyperbolic diffeomorphisms whose central direction is non-uniformly contracting (negative Lyapunov exponents) almost everywhere have the Bernoulli property: the system is equivalent to an i. i. d. (independently identically distributed) random process. In particular, these systems are mixing: correlations of integrable functions go to zero as time goes to infinity. We also extend this result in two different ways. Firstly, for 3-dimensional diffeomorphisms, if one requires only non-zero (instead of negative) Lyapunov exponents then one still gets a quasi-Bernoulli property. Secondly, if one assumes accessibility (any two points are joined by some path whose legs are stable segments and unstable segments) then it suffices to requires the mostly contracting property on a positive measure subset, to obtain the same conclusions.     

Localization and Relaxation in Anisotropically Disordered Systems

We discuss localization and the scattering of excitations in bifractals, a model of anisotropically disordered systems. The localization behavior is anisotropic. With the increase of energy, the excitation crosses over from an extended wave to a wave extended in one subspace while localized in another, then to a wholly-localized wave. The loffe-Regel frequency is shown to be in the wholly-localized regime. Relaxation processes are calculated for the emission and absorption of localized vibrational excitations by a localized electronic state.The anisotropy makes effects on the results.