A comment on Anderson's transition in systems with purely off-diagonal disorder

We make a comment on a recent paper by Antoniou and Economou concerning the electron localization in systems with off-diagonal randomness. Particularly, we discuss the validity of the ordinary cluster coherent potential approximation and of the L(E) method, both of which were used as grounds to draw their conclusion.     

Diagrammatic approach to the off-diagonal randomness in the coherent potential approximation

The off-diagonal randomness in the coherent potential approximation (CPA) is self-consistently treated by extending Nickel and Krumhansl's diagram-resummation technique CPA(2). New diagrams are introduced to represent the off-diagonal potentials and multiple occupancy correction is made in a completely self-consistent way.     

Numerical results for Anderson localisation in the presence of off-diagonal disorder

Numerical calculations of the inverse participation ratio have been performed for the diamond cubic lattice with nearest neighbour interactions, and both diagonal and off-diagonal disorder. We confirm the prediction of Economou and Antoniou that off-diagonal disorder cannot, per se, produce an Anderson transition, although it reduces the amount of diagonal disorder necessary for the transition. In the presence of off-diagonal disorder alone we do not detect any substantial localisation even in the wings of the band.     

Effective-medium theory for goldstone systems with off-diagonal disorder

An effective medium theory for low temperature phonons and other Goldstone excitations with diagonal and off-diagonal disorder is developed which preserves the herglotz property of the self-energy and is correct in both dilute limits c→0 and c→1.     

Coherent potential approximation for a disordered diatomic linear chain having diagonal and off-diagonal randomness

A coherent potential approximation (cpa) for a mixed diatomic linear chain including both mass and force constant changes has been developed. In this case an impurity atom substituted at a particular site in one of the sublattices couples with two nearest neighbour atoms in the other sublattices. The diatomic linear chain is therefore considered as a tetratomic linear chain, the size of the unit cell being twice the original. Thecpa density of states and the dielectric susceptibility have been calculated. The numerical values of the later have been calculated in theata (averaget-matrix approximation) limit. Comparison of these results with the experimental and other computer calculations show a qualitative agreement.     

Anderson localization in different one-dimensional systems with off-diagonal disorder and spin-dependence

The localization properties of certain spin-dependent, one-dimensional electronic systems with only off-diagonal disorder are studied. In higher dimensions (d=2,3) the models considered would correspond to different universality classes, whereas ford=1 no qualitative difference is found: ForE=0, all eigenstates are exponentially localized, whereas forE0 the localization length diverges logarithmically, such that exactly atE=0 the geometric average of the transmission coefficient would decay with increasing chain lengthL as exp (-const. ·L ^1/2), instead of the usual, exponential decay.ForE=0, in the interior of the band, the localization lengthr ₀ diverges W ₂ ^–2 in the limit of weak disorder (W ₂0), whereas just at the band edge one has roughlyr ₀W ₂ ^–2/3. A universal recursion relation, depending only on the energy and on certain randomly distributed determinants, determines the localization length and the density of states for all systems considered.     

Limits of infinite interaction radius,dimensionality and the number of components for random operators with off-diagonal randomness

We consider random one-body operators that are analogs of the statistical mechanics Hamiltonians with a varying interaction radiusR, the dimensionality of spaced and the number of the field components (orbitals)n. We prove that all the moments of the Green functions for nonreal energies of these operators converge asR, d, n to the products of the average Green functions, just as in the mean field approximation of statistical mechanics. We find in particular the selfconsistent equation for the limiting integrated density of states and the limiting form of the conductivity, which is nonzero on the whole support of the integrated density of states.     

On intrinsic randomness of dynamical systems

We discuss the problem of nonunitary equivalence, via positivity-preserving similarity transformations, between the unitary groups associated with deterministic dynamical evolution and semigroups associated with stochastic processes. Dynamical systems admitting such nonunitary equivalence with stochastic Markov processes are said to beintrinsically random. In a previous work, it was found that the so-called Bernoulli systems (discrete time) are intrinsically random in this sense. This result is extended here by showing that a more general class of dynamical systems—the so-calledK systems andK flows—are intrinsically random. The connection of intrinsic randomness with local instability of motion is briefly discussed. We also show that Markov processes associated through nonunitary equivalence tononisomorphic K flows are necessarily non-isomorphic.Dr. Goldstein's research was supported in part by NSF Grant No. PHY78-03816.     

Mode-coupling theory for purely diffusive systems

A mode-coupling formalism is developed for multicomponent systems of particles performing diffusive motion in a uniform host medium. The mode-coupling equations are derived from a set of nonlinear fluctuating diffusion equations by expanding the concentration-dependent diffusion constants about their equilibrium values. From the mode-coupling equations the dominant long time behavior of current-current and super-Burnett correlation functions is derived. As specific applications I consider the long time behaviors of these correlation functions for collective and tracer diffusion in a one-component lattice gas with particle-conserving stochastic dynamics. The results agree with those from exactly solvable models and computer simulations.     

Statistics of amplified light in periodically correlated layered systems with randomness

Abstract

We study the statistics of reflection and transmission coefficients of light in randomly layered amplifying media that are periodic on average. We are interested in one-dimensional universal scaling behaviours in such systems. Our study shows that while a homogeneous medium boundary condition is capable of reproducing universal scaling at low frequencies, a periodic medium boundary condition is necessary for high frequencies. Although the statistics depends on the boundary condition, the saturation length, where the reflection coefficient reaches a stationary distribution, and the localization length do not. Implications of these results are discussed.     

Near band centre states for one-dimensional off-diagonal disordered systems

We study and characterize the eigenstates near the centre of the band of a one-dimensional tight binding model with off-diagonal disorderW _T . We report a new exponent for the localization length on an energy-dependent range of disorderW _T . We correlate this feature with a change of structure of the wave-function displayed by the behaviour of its density-density correlation function.     

Intrinsic randomness and spontaneous symmetry-breaking in explosive systems

The effect of fluctuations of either thermodynamic or environmental origin on ignition in explosive systems is analyzed, with special emphasis on thermal explosion. A simple model due to Semenov is first analyzed in the zero-dimensional approximation. It is shown that the ignition times exhibit a wide dispersion, which at the level of the probability distribution of temperature shows up as a transient bimodality. Next, an extension to a spatially distributed system is developed. It is shown that fluctuations induce unexpected symmetry-breaking phenomena, reflected by a considerable dispersion of the position of the first hot spot initiated in the system.     

On spin systems with quenched randomness: Classical and quantum

The rounding of first-order phase transitions by quenched randomness is stated in a form which is applicable to both classical and quantum systems: The free energy, as well as the ground state energy, of a spin system on a d-dimensional lattice is continuously differentiable with respect to any parameter in the Hamiltonian to which some randomness has been added when d≤2. This implies absence of jumps in the associated order parameter, e.g., the magnetization in the case of a random magnetic field. A similar result applies in cases of continuous symmetry breaking for d≤4. Some questions concerning the behavior of related order parameters in such random systems are discussed.     

Suppression of ground-state magnetization in finite-size systems due to off-diagonal interaction fluctuations

We study a generic model of interacting fermions in a finite-size disordered system. We show that the off-diagonal interaction matrix elements induce density of state fluctuations which generically favor a minimum spin ground state at large interaction amplitude, U. This effect competes with the exchange effect which favors large magnetization at large U, and it suppresses this exchange magnetization in a large parameter range. When off-diagonal fluctuations dominate, the model predicts a spin gap which is larger for odd-spin ground states as for even spin, suggesting a simple experimental signature of this off-diagonal effect in Coulomb blockade transport measurements.     

Density of states in coupled chains with off-diagonal disorder

We compute the density of states rho(varepsilon) in N coupled chains with random hopping. At zero energy, rho(varepsilon) shows a singularity that strongly depends on the parity of N. For odd N, rho(varepsilon) approximately 1/|varepsilonln (3)varepsilon|, with and without time-reversal symmetry. For even N, rho(varepsilon) approximately |lnvarepsilon| in the presence of time-reversal symmetry, while there is a pseudogap, rho(varepsilon) approximately |varepsilonlnvarepsilon|, in the absence of time-reversal symmetry.     

Measuring randomness with periodic media

We show that the sensitivity to disorder of certain physical properties of periodic media depends on whether the disorder is truly random or not. This allows to utilize periodic media for testing sequences of random numbers and to quantify their departure from true randomness via simple transmission and/or reflection data analysis. This physics-based model shows promises for device applications to test random data.     

Multifractal analysis and randomness of almost periodic forced two-level systems

Numerical results of multifractal analysis of the (quantum) dynamics of forced two-level systems under some almost periodic time dependence are reported. The aims are to check the presence of multifractal characteristics of these systems, and also to inspect if different degrees of aperiodicities in the force are transferred to the system dynamics. Although the dynamics present signatures of the randomness of the perturbation, it does not follow strictly its autocorrelation type. The wavelet transform is the tool used to carry out the multifractal analysis.