Percolation techniques in disordered spin flip dynamics: Relation to the unique invariant measure

We consider lattice spin systems with short range but random and unbounded interactions. We give criteria for ergodicity of spin flip dynamics and estimate the speed of convergence to the unique invariant measure. We find for this convergence a stretched exponential in time for a class of directed dynamics (such as in the disordered Toom or Stavskaya model). For the general case, we show that the relaxation is faster than any power in time. No assumptions of reversibility are made. The methods are based on relating the problem to an oriented percolation problem (contact process) and (for the general case) using a slightly modified version of the multiscale analysis of e.g. Klein (1993).     

Hopping conduction and bacteria: transport in disordered reaction-diffusion systems

We report some basic results regarding transport in disordered reaction-diffusion systems with birth (A-->2A), death (A-->0), and binary competition (2A-->A) processes. We consider a model in which the growth process is only allowed to take place in certain areas--"oases"--while the rest of space--the "desert"--is hostile to growth. In the limit of low oasis density, transport is mediated through rare "hopping" events, necessitating the inclusion of discreteness effects in the model. By first considering transport between two oases, we are able to derive an approximate expression for the average time taken for a population to traverse a disordered medium.     

Nanoscale phase coexistence and percolative quantum transport

We study the nanoscale phase coexistence of ferromagnetic metallic and antiferromagnetic insulating (AFI) regions by including the effect of AF superexchange and weak disorder in the double exchange model. We use a new Monte Carlo technique, mapping on the disordered spin-fermion problem to an effective short range spin model, with self-consistently computed exchange constants. We recover "cluster coexistence" as seen earlier in exact simulation of small systems. The much larger sizes, approximately 32 x 32, accessible with our technique, allow us to study the cluster pattern for varying electron density, disorder, and temperature. We track the magnetic structure, obtain the density of states, with its "pseudogap" features, and, for the first time, provide a fully microscopic estimate of the resistivity in a phase coexistence regime, comparing it with the "percolation" scenario.     

Inhomogeneous ferromagnetism and unconventional charge dynamics in disordered double exchange magnets

We solve the double exchange model in the presence of arbitrary substitutional disorder by using a self-consistently generated effective Hamiltonian for the spin degrees of freedom. The magnetic properties are studied through classical Monte Carlo while the effective exchange, D(ij), is calculated by solving the disordered fermion problem, and renormalized self-consistently with increasing temperature. We present results on the conductivity, magnetoresistance, optical response, and "real space" structure of the inhomogeneous ferromagnetic state, and compare our results with charge dynamics in disordered La1-xSrxMnO3. The large sizes, O(10(3)), accessible within our method allows a complete, controlled calculation on the disordered strongly interacting problem.     

Adjoint-based optimization of PDEs in moving domains

In this investigation we address the problem of adjoint-based optimization of PDE systems in moving domains. As an example we consider the one-dimensional heat equation with prescribed boundary temperatures and heat fluxes. We discuss two methods of deriving an adjoint system necessary to obtain a gradient of a cost functional. In the first approach we derive the adjoint system after mapping the problem to a fixed domain, whereas in the second approach we derive the adjoint directly in the moving domain by employing methods of the noncylindrical calculus. We show that the operations of transforming the system from a variable to a fixed domain and deriving the adjoint do not commute and that, while the gradient information contained in both systems is the same, the second approach results in an adjoint problem with a simpler structure which is therefore easier to implement numerically. This approach is then used to solve a moving boundary optimization problem for our model system.     

Nonsingular global string compactifications

We consider an exotic "compactification" of spacetime in which there are two infinite extra dimensions, using a global string instead of a domain wall. By having a negative cosmological constant we prove the existence of a nonsingular static solution using a dynamical systems argument. A nonsingular solution also exists in the absence of a cosmological constant with a time-dependent metric. We compare and contrast this solution with the Randall-Sundrum universe and the Cohen-Kaplan spacetime and consider the options of using such a model as a realistic resolution of the hierarchy problem.     

Field-dependent charge-carrier trapping process in disordered systems

We study the diffusive spreading of excees carriers in trapping systems to which a bias electric field is applied. Since for the intermediate time range the present model appears to be identical to the electronic disordered system (the Anderson model), we have been able to derive the time and field dependencies of carrier survival probability. The results of the exact calculations obtained might provide significant insight into the transport phenomena occurring in disordered systems. It is shown that in the long-time limit the coherent potential approximation can serve as an exact solution of the problem.     

Random channel kinetics for reaction-diffusion systems

A random channel approach is developed for reaction-diffusion processes in disordered systems. Although the starting point of our research is the kinetic study of the decay and preservation of marine organic carbon, our approach can be used for describing other disordered kinetic catalytic processes with random pathways. We consider a generic catalytic mechanism with two species: (a) a catalyst, which is continuously produced by a variable number of independent sources randomly distributed in space; this catalyst diffuses from the sources and is degrading according to a first order kinetic law; the generation, the degradation and the diffusion of the catalyst balance each other out and a stationary concentration field is generated; (b) an active species, which decays according to a second order kinetic law; the decay rate is proportional to the product of the concentrations of the catalyst and the concentration of the active species. We show that the catalyst concentration field can be represented by the sum of a random number of Yukawa-like potentials. The average value of the survival function of the active species can be expressed as a grand canonical average of a nonlinear functional of the catalyst field and can be evaluated exactly. We show that a good approximation is given by a nearest neighbor approach, where only the contribution of the closest source is taken into account for the computation of the random concentration field of the catalyst. We discuss the application of the model to the problem of decay and preservation of marine organic carbon. With minor adaptation the model can be applied to other problems of disordered kinetics, such as spatially distributed heterogeneous catalytic processes.     

Periodically driven ergodic and many-body localized quantum systems

We study dynamics of isolated quantum many-body systems whose Hamiltonian is switched between two different operators periodically in time. The eigenvalue problem of the associated Floquet operator maps onto an effective hopping problem. Using the effective model, we establish conditions on the spectral properties of the two Hamiltonians for the system to localize in energy space. We find that ergodic systems always delocalize in energy space and heat up to infinite temperature, for both local and global driving. In contrast, many-body localized systems with quenched disorder remain localized at finite energy. We support our conclusions by numerical simulations of disordered spin chains. We argue that our results hold for general driving protocols, and discuss their experimental implications.     

Regularization of diagrammatic series with zero convergence radius

The divergence of perturbative expansions which occurs for the vast majority of macroscopic systems and follows from Dyson's collapse argument prevents the direct use of Feynman's diagrammatic technique for controllable studies of strongly interacting systems. We show how the problem of divergence can be solved by replacing the original model with a convergent sequence of successive approximations which have a convergent perturbative series while maintaining the diagrammatic structure. As an instructive model, we consider the zero-dimensional |ψ|? theory.     

Path degeneracy and EXAFS analysis of disordered materials

Analysis of EXAFS data measured on a material with a disordered local configuration environment around the absorbing atom can be challenging owing to the proliferation of photoelectron scattering paths that must be considered in the analysis. In the case where the absorbing atom exists in multiple inequivalent sites, the problem is compounded by having to consider each site separately. A method is proposed for automating the calculation of theory for inequivalent sites, then averaging the contributions from sufficiently similar scattering paths. With this approach, the complexity of implementing a successful fitting model on a highly disordered sample is reduced. As an example, an analysis of Ti K‐edge data on zirconolite, CaZrTi₂O₇, which has three inequivalent Ti sites, is presented.     

DNA unzipping and the unbinding of directed polymers in a random media

We consider the unbinding of a directed polymer in a random media from a wall in d=1+1 dimensions and a simple one-dimensional model for DNA unzipping. Using the replica trick we show that the restricted partition functions of these problems are identical up to an overall normalization factor. Our finding gives an example of a generalization of the stochastic matrix form decomposition to disordered systems, a method which allows us to reduce the dimensionality of the problem. The equivalence between the two problems, for example, allows us to derive the probability distribution for finding the directed polymer a distance z from the wall. We discuss implications of these results for the related Kardar-Parisi-Zhang equation and the asymmetric exclusion process.     

Universality of “level curvature” distribution for large random matrices: systematic analytical approaches

We perform an extensive analytical study of distributions of “level curvatures” (the second derivatives of eigenvalues with respect to a perturbation parameter) for different classers of random matrice. First, we consider the case of three Gaussian ensembles: GUE, GOE and GSE. This part of our calculation is complementary to that done recently by von Oppen [22, 23], but evaluation goes along different lines and allows to treat all the three cases uniformly. In the second part of the paper we exploit completely another method allowing to treat the problem analytically for the broad class of disordered systems subject to time-reversal symmetry breaking perturbation. That gives us a possibility to prove the conjecture by Zakrzewski and Delande [17] for the ensemble of symmetric sparse random matrices.     

A functional integral approach to shock wave solutions of Euler equations with spherical symmetry

Forn×n systems of conservation laws in one dimension without source terms, the existence of global weak solutions was proved by Glimm [1]. Glimm constructed approximate solutions using a difference scheme by solving a class of Riemann problems.In this paper, we consider the Cauchy problem for the Euler equations in the spherically symmetric case when the initial data are small perturbations of the trivial solution, i.e.,u0 and constant, whereu is velocity and is density. We show that this Cauchy problem can be reduced to an ideal nonlinear problem approximately. If we assume all the waves move at constant speeds in the ideal problem, by using Glimm's scheme and an integral approach to sum the contributions of the reflected waves that correspond to each path through the solution, we get uniform bounds on theL norm and total variational norm of the solutions for all time. The geometric effects of spherical symmetry leads to a non-integrable source term in the Euler equations. Correspondingly, we consider an infinite reflection problem and solve it by considering the cancellations between reflections of different orders in our ideal problem. Thus we view this as an analysis of the interaction effects at the quadratic level in a nonlinear model problem for the Euler equations. Although it is far more difficult to obtain estimates in the exact solutions of the Euler equations due to the problem of controlling the time at which the cancellations occur, we believe that this analysis of the wave behaviour will be the first step in solving the problem of existence of global weak solutions for the spherically symmetric Euler equations outside of fixed ball.     

A super-Ohmic energy absorption in driven quantum chaotic systems

We consider energy absorption by driven chaotic systems of the symplectic symmetry class. According to our analytical perturbative calculation, at the initial stage of evolution the energy growth with time can be faster than linear. This appears to be an analog of weak anti-localization in disordered systems with spin-orbit interaction. Our analytical result is also confirmed by numerical calculations for the symplectic quantum kicked rotor.     

Anderson-Mott transition driven by spin disorder: spin glass transition and magnetotransport in amorphous GdSi

A zero temperature Anderson-Mott transition driven by spin disorder can be "tuned" by an applied magnetic field to achieve colossal magnetoconductance. Usually this is not possible since spin disorder by itself cannot localize a high density electron system. However, the presence of strong structural disorder can realize this situation, self-consistently generating a disordered magnetic ground state. We explore such a model, constructed to understand amorphous GdSi, and highlight the emergence of a spin glass phase, Anderson-Mott signatures in transport and tunneling spectra, and unusual magneto-optical conductivity. We solve a disordered strong coupling fermion-spin-lattice problem essentially exactly on finite systems and account for all the qualitative features observed in magnetism, transport, and the optical spectra in this system.     

Step Density Profiles in Localized Chains

We consider two types of strongly disordered one-dimensional Hamiltonian systems coupled to baths (energy or particle reservoirs) at the boundaries: strongly disordered quantum spin chains and disordered classical harmonic oscillators. These systems are believed to exhibit localization, implying in particular that the conductivity decays exponentially in the chain length L. We ask however for the profile of the (very slowly) transported quantity in the steady state. We find that this profile is a step-function, jumping in the middle of the chain from the value set by the left bath to the value set by the right bath. This is confirmed by numerics on a disordered quantum spin chain of 9 spins and on much longer chains of harmonic oscillators. From theoretical arguments, we find that the width of the step grows not faster than \(\sqrt{L}\), and we confirm this numerically for harmonic oscillators. In this case, we also observe a drastic breakdown of local equilibrium at the step, resulting in a heavily oscillating temperature profile.     

On the Boyd-Kadomstev System for a Three-Wave Coupling Problem and its Asymptotic Limit

We consider the Boyd-Kadomstev system which is in particular a model for the Brillouin backscattering in laser-plasma interaction. It couples the propagation of two laser beams, the incoming and the backscattered waves, with an ion acoustic wave which propagates at a much slower speed. The ratio ${\varepsilon}$ between the plasma sound velocity and the (group) velocity of light is small, with typical value of order 10^?3. In this paper, we make a rigorous analysis of the behavior of solutions as ${\varepsilon \to 0}$ . This problem can be cast in the general framework of fast singular limits for hyperbolic systems. The main new point which is addressed in our analysis is that the singular relaxation term present in the equation is a nonlinear first order system.