Th. M. Nieuwenhuizen

Singular behavior of the density of states and the Lyapunov coefficient in binary random harmonic chains

We study the integrated density of statesH(ω2) of a chain of harmonic oscillators with a binary random distribution of the masses. We show in particular that there is a dense set of values of the squared frequency for which the differenceH(ω2+ɛ)-H(ω2) has a singularity of the type ¦ɛ¦2α, multiplied by a periodic function of ln ¦ɛ¦, where the exponent α and the period depend continuously onω2. In the region where α < 1/2,H is not differentiate on a dense set of points. The same type of singularities is also present in the Lyapunov coefficient.

Exact critical behavior of two-dimensional wetting problems with quenched disorder

The wetting transition in the presence of a random substrate is studied in two dimensions, using a restricted solid-on-solid model. The singular part of the quenched free energy and specific heat is calculated exactly by means of the replica trick. Disorder introduces logarithmic corrections to the results of the pure system. The divergent part of the width of the wetting layer is also evaluated: here no corrections to the pure case are obtained. The method employed uses a field-theoretic calculation (in terms of Goldstone diagrams) of the ground-state energy of an effective many-body Hamiltonian. The validity of the replica method is tested numerically.

A Soluble Quasi-Crystalline Magnetic Model: The XY Quantum Spin Chain

We consider the quantum XY spin chain with quasi-periodic two-valued exchange couplings and a uniform transverse field. The equivalence of the system with a free fermionic model permits a detailed analysis of its thermodynamics. The zero-temperature magnetization is a Cantor function of the applied field. The zero-field specific heat and susceptibility have a power-law behaviour at low temperature, modulated by periodic amplitudes. The critical exponents interpolate continuously between those of the periodic and of the random model.

Excess noise in a hopping model for a resistor with quenched disorder

We consider a model for independent charged particles, hopping on a lattice with static disorder in the waiting times. The excess current noise is calculated and shown to be related to resistance noise and arising from mobility fluctuations. It is also related to the four point super-Burnett-function. The strength of the noise is calculated at small frequencies for weak disorder (classical long time tails) and for strong disorder, when it may behave like I/f. In that case the Hooge factor equals the fraction of deep trapping centers.

Lifshitz tails and long-time decay in random systems with arbitrary disorder

In random systems, the density of states of various linear problems, such as phonons, tight-binding electrons, or diffusion in a medium with traps, exhibits an exponentially small Liftshitz tail at band edges. When the distribution of the appropriate random variables (atomic masses, site energies, trap depths) has a delta function at its lower (upper) bound, the Lifshitz singularities are pure exponentials. We study in a quantitative way how these singularities are affected by a universal logarithmic correction for continuous distributions starting with a power law. We derive an asymptotic expansion of the Lifshitz tail to all orders in this logarithmic variable. For distributions starting with an essential singularity, the exponent of the Lifshitz singularity itself is modified. These results are obtained in the example of harmonic chains with random masses. It is argued that analogous results hoid in higher dimensions. Their implications for other models, such as the long-time decay in trapping problems, are also discussed.

Diffusion and survival in a medium with imperfect traps

This paper deals with independent particles diffusing on a line with traps at random positions. It is shown how the long-time decay of the survival probability is exhanced when particles do not necessarily disappear upon hitting a trap. The results are compared with predictions for a model where particles are either absorbed or reflected by traps.

Singularities in Spectra of Disordered Systems: An Instanton Approach for Arbitrary Dimension and Randomness

The spectra of random systems usually have an exponential singularity (Lifshitz tail) in the density of states at band edges: this singularity is a pure exponential for, e.g., the Anderson model with a binary potential distribution. It has been shown recently, at least in one dimension, that the Lifshitz singularity is affected by a universal logarithmic correction in the case of, e.g., uniform distributions. The precise form of the Lifshitz tail is derived here, by means of a field-theoretic description, and of instanton calculus. This very general scheme provides new results for an arbitrary distribution of potentials, in arbitrary dimension.

Lifshitz singularities in random harmonic chains: Periodic amplitudes near the band edge and near special frequencies

We give a complete description of the scaling behavior of the integrated density of states of random harmonic chains with random masses near the band edgeωmax and near special frequenciesωs. There are four different situations:ω ↑ωmax,ω ↓ωs,ω ↑ωs (critical case),ω ↑ωs (general case). Our analytic results have the form of infinite sums involving Fourier coefficients of the scaling behavior of the Dyson-Schmidt functionat the special frequency or the band edge. Binary mass distributions are considered in detail in the limit of a small fractionp of light masses. Our predictions are compared with extensive numerical data.

Thermally Activated Flux Creep in High-Temperature Superconductors: a Stochastic Model

A model is presented which describes magnetic properties of high-Tc superconductors when flux creep is the leading mechanism for dissipation. This model reduces the behaviour of a complex current pattern to the behaviour of essentially one single current loop which contains a fluctuating number of flux quanta. It is shown that the dynamics of such a system describes widely observed phenomena such as logarithmic decay of the magnetization, broadening of the resistive transition as a function of an external field and nonlinearity in the a.c.-susceptibility. Away from equilibrium, fluctuations in the magnetization and even harmonics in the a.c.-susceptibility decay algebraically in time.

Dynamical properties of 2D systems with site disorder

Dynamical properties are studied of particles which perform random walks on square lattices, where a fraction c of the sites has been replaced by impurities. The latter may be either better conducting than the host, or worse conducting or even non-conducting. The quantities studied are the velocity autocorelation function, the return probability, the density of states and Burnett functions. The results include short-time, finite-time and long-time behavior. Prefactors of the long-time behavior are calculated up to second order in c. Comparison with numerical simulations is made. Different definitions of the diffusion coefficient (via the second moment of displacement, the return probability or the density of states) are shown to give different answers.