Cumulant approach to dynamical correlation functions

A new theoretical approach, based on the introduction of cumulants, to calculate time dependent correlation functions at zero temperature is presented. In contrast to usual diagrammatic treatments of Green functions, the present method can be applied to operators which do not satisfy standard fermion or boson commutation relations. The introduction of cumulants is equivalent to applying the linked-cluster theorem. Thus, only connected expectation values have to be evaluated, even in cases for which Wick's theorem does not apply. The method expresses correlation functions in a form accessible to projection technique approaches. However, it circumvents a conceptual difficulty inherent in standard projection technique. There static expectation values, which are defined within the exact ground state, have to be evaluated independently. Our technique is a generalization, to dynamical aspects, of a recently introduced cumulant approach, which was restricted to the calculation of static ground-state properties. It seems to be an appropriate theoretical tool to treat the influence of strong electronic correlations like, e.g., in the new high-T _c superconducting materials. Two applications of the formalism have been given recently.     

Cumulant approach to dynamical correlation functions at finite temperatures

A new theoretical approach, based on the introduction of cumulants, to calculate thermodynamic averages and dynamical correlation functions at finite temperatures is developed. The method is formulated in Liouville instead of Hilbert space and can be applied to operators which do not require to satisfy fermion or boson commutation relations. The application of the partitioning and projection methods for the dynamical correlation functions is considered. The present method can be applied to weakly as well as to strongly correlated systems.     

Time-dependent correlation functions for random walks on bond disordered cubic lattices

For hopping models on cubic lattices with a fractionc of impurity bonds, time-dependent transport properties and correlation functions (long-time tails) are calculated through a systematicc-expansion (in the percolation literature referred to as high-density expansion), using a method developed in an earlier paper. The time-dependent diffusion coefficient, velocity autocorrelation function (VACF), and Burnett functions are calculated exact toO(c) for allt, and exact toO(c ₂ ) for long times only. A comparison is made with the results of the effective medium approximation, and numerical results are given for the square lattice.     

Complexity of random energy landscapes, glass transition, and absolute value of the spectral determinant of random matrices

Finding the mean of the total number N(tot) of stationary points for N-dimensional random energy landscapes is reduced to averaging the absolute value of the characteristic polynomial of the corresponding Hessian. For any finite N we provide the exact solution to the problem for a class of landscapes corresponding to the "toy model" of manifolds in a random environment. For N>1 our asymptotic analysis reveals a phase transition at some critical value mu(c) of a control parameter mu from a phase with a finite landscape complexity: N(tot) approximately e(N Sigma), Sigma(mu0 to the phase with vanishing complexity: Sigma(mu>mu(c))=0. Finally, we discuss a method of dealing with the modulus of the spectral determinant applicable to a broad class of problems.     

Mean relaxation time approximation for dynamical correlation functions in stochastic systems near instabilities

The linewidth factor of the order parameter autocorrelation function for the mean fieldn-vector Ginzburg-Landau model is determined numerically forn>2. This generalizes results for the single mode laser (n=2) obtained in Part II [Nadler, W., Schulten, K.: Z. Phys. B—Condensed Matter].     

Mean relaxation time approximation for dynamical correlation functions in stochastic systems near instabilities

We present a simple approximation for dynamical correlation functions in stochastic systems which reproduces the high as well as the low frequency behaviour of the exact correlation functions. The approximation is applied in its lowest order to diffusion in a quartic potential and to autocatalytic chemical reaction systems as described by the Schlögl models. The results are compared to those from the conventional Mori-Zwanzig projection operator approach which reproduces only the short-time relaxation of the systems considered. The new approximation describes correctly slow relaxation processes, e.g. barrier crossing in a quartic potential and the slowing down of dynamic processes in finite autocatalytic systems near first and second order transitions.     

Exact expressions for row correlation functions in the isotropicd=2 ising model

We present exact explicit expressions for the row spin-spin correlation functions _{00 n}0 in the isotropicd= 2 Ising model, in terms of elliptic integrals, forn 5. We also give a general structural formula for _{00 n}0.     

Intermediate asymptotic expressions for high-order spin correlation functions in a two-dimensional classical ferromagnet

Different-point spin correlation functions are calculated for a two-dimensional classical ferromagnet in a pacerturbative range of distances r: a<r?m ^?1, where a is the lattice parameter and m ^?1 is the correlation length. The expressions for the four-and higher-order correlation functions are presented.     

Epidemic prevalence on random mobile dynamical networks: individual heterogeneity and correlation

In this paper, we extend the susceptible-infected-susceptible (SIS) epidemiological model on a random dynamical network composed of mobile individuals, in which the infection is caused by the collisions between susceptible and infected individuals at the spreading rate proportional to their susceptibilities and infectivities. We analytically study the criticality of spreading dynamics under different distributions of individual susceptibility and infectivity, and numerically verify the cases of power-law and (or) Gaussian distributions. Our findings show that the heterogeneity of individual susceptibility and infectivity increases the epidemic threshold, and the positive correlation of individual susceptibility and infectivity avails to the epidemic prevalence.     

Computation of dynamical correlation functions of heisenberg chains in a magnetic field

We compute the momentum- and frequency-dependent longitudinal spin structure factor for the spin-1/2 XXZ Heisenberg spin chain in a magnetic field, using exact determinant representations for form factors on the lattice. Multiparticle (i.e., multispinon) contributions are computed numerically throughout the Brillouin zone, yielding saturation of the sum rule to high precision.     

Finite-N fluctuation formulas for random matrices

For the Gaussian and Laguerre random matrix ensembles, the probability density function (p.d.f.) for the linear statistic Σ _j ^N =1 (x _j ? 〈x〉) is computed exactly and shown to satisfy a central limit theorem asN → ∞. For the circular random matrix ensemble the p.d.f.’s for the statistics ½Σ _j ^N =1 (θ _j ?π) and ? Σ _j ^N =1 log 2 |sinθ _j/2| are calculated exactly by using a constant term identity from the theory of the Selberg integral, and are also shown to satisfy a central limit theorem asN → ∞.     

A formula for the determinant of a sum of matrices

We give a formula, involving circular words and symmetric functions of the eigenvalues, for the determinant of a sum of matrices. Theorem of Hamilton-Cayley is deduced from this formula.UQAM and LITPUniversité Paris 7 and LITP.     

Correlations for parameter— dependent random matrices

Correlations for parameter-dependent Gaussian random matrices, intermediate between symmetric and Hermitian and antisymmetric Hermitian and Hermitian, are calculated. The (dynamical) density-density correlation between eigenvalues at different values of the parameter is calculated for the symmetric to Hermitian transition and the scaledN→∞ limit is computed. For the antisymmetric Hermitian to Hermitian transition the equal-parametern-point distribution function is calculated and the scaled limit computed. A circular version of the antisymmetric Hermitian to Hermitian transition is formulated. In the thermodynamic limit the equal-parameter distribution function is shown to coincide with the scaled-limit expression of this distribution for the Gaussian antisymmetric Hermitian to Hermitian transition. Furthermore, the thermodynamic limit of the corresponding density-density correlation is computed. The results for the correlations are illustrated by comparison with empirical correlations calculated from numerical data obtained from computer-generated Gaussian random matrices.     

Universal dynamical control of local decoherence for multipartite and multilevel systems

A unified theory is given of dynamically modified decay and decoherence of field-driven multilevel multipartite entangled states that are weakly coupled to zero-temperature baths or undergo random phase fluctuations. The theory allows for arbitrary local differences in their coupling to the environment. Due to such differences, the optimal driving-field modulation to ensure maximal fidelity is found to substantially differ from conventional “Bang-Bang” or π-phase flips of the single-qubit evolution.     

Inverse spectral theory for random Jacobi matrices

We give necessary and sufficient conditions for a Herglotz function to be thew-function of a random stationary Jacobi matrix.This paper is dedicated to the memory of Mark Kac.