One-dimensional Random Systems Research Articles

Classical Hopping Conduction in Random One-Dimensional Systems: Nonuniversal Limit Theorems and Quasilocalization Effects

The $L\ensuremath{\rightarrow}\ensuremath{\infty}$ asymptotic properties of ${\ensuremath{\rho}}_{L}(g)$, the probability distribution of the classical hopping conductivity ${g}_{L}$ corresponding to random one-dimensional systems of length $L$, are determined. These properties are nonuniversal, and become anomalous if the probability density $\ensuremath{\rho}(w)$ of the random near-neighbor hopping rates is such that $\ensuremath{\int}{0}^{\ensuremath{\infty}}\mathrm{dw}\ensuremath{\rho}(w){w}^{\ensuremath{-}1}$ does not exist. The associated quasilocalization effects are discussed and their experimental observability is speculated upon.

Effect of correlations on the low-frequency behavior of random one-dimensional systems

We study the motion of a particle or elementary excitation that hops between nearest neighbors in a one-dimensional array of traps. The trap depths and transition probabilities are random, but they may show correlations over short distances. When certain averages of the inverse transition rates exist, the low-lying eigenvalues of the master equation for the system are distributed like those of some equivalent regular chain. However, the nature of this equivalent regular chain depends on the degree of correlation of the random transition rates. Moreover, for a chain with periodic boundary conditions the degeneracy of the eigenvalues in the regular chain is lifted. The amount of this splitting depends on the degree of randomness and on the chain length. The density of low-lying eigenvalues is closely related to the asymptotic form of the probability for finding the particle at some large time at the same place where it started its random walk. Our results are obtained by studying the moments of the reciprocals of the eigenvalues, in which the distribution of low-lying eigenvalues leaves a characteristic fingerprint.

Excitation dynamics in random one-dimensional systems

In a number of recent publications, [1] – [5], we have discussed the asymptotic form of the dynamics of a general type of random one-dimensional chains. The equations we discuss are of the form $${{\rm{C}}_n}\left( {\frac{{{\rm{d}}{{\rm{V}}_n}}}{{{\rm{dt}}}}} \right) = {{\rm{W}}_{n + 1}}\left( {{{\rm{V}}_{n,n + 1}} - {{\rm{V}}_n}} \right) + {{\rm{W}}_{n - 1}}\left( {{{\rm{V}}_{n,n - 1}} - {{\rm{V}}_n}} \right),\,\,\,\,{{\rm{W}}_{n,n + 1}} = {{\rm{W}}_{n,n + 1}},$$ (1) are independent positive random variables. Equations of this type arise in a variety of physical contexts. They can represent a master equation for hopping-type transport over random barriers (the Cn=1, the Wn,n+1 random hopping rates); a master equation for excitation transfer along a one-dimensional array of traps of random depth (the Cn random Boltzmann factors, the Wn,n+1=1); an electric transmission line (the Cn random capacitors or the Wn,n+1 random conductances); a random Heisenberg ferromagnetic chain at low temperatures (the Cn=1, Vn representing spin wave destruction operators, and the Wn,n+1 random near-neighbor exchange integrals). Replacing dVn/dt in (1) by d2Vn/dt2, they represent a harmonic chain with random masses (the Cn) or random force constants (the Wn,n+1).

Some comments on hopping in random one-dimensional systems

For a general type of one-dimensional disordered lattice systems we summarize and discuss the present status of the theoretical investigations concerning the low-frequency (long-time) asymptotic behavior of different quantities. This is done to clarify certain points raised by the computer-simulation results of Richards and Renken.

Low-energy density of states and long-time behaviour of random chanins: a scaling approach

The asymptotic low-energy and long-time properties of random one-dimensional systems with harmonic chain-type eigenvalue equations are discussed. For a general class of random distributions of the coupling constants W, rho (W), an inherent length scale can be defined, and the corresponding systems exhibit conventional (universal) asymptotic properties. If rho (W) is such that no intrinsic length is defined, a unique energy-dependent localisation length can be determined which leads to anomalous (non-universal) diffusion and density-of-states indices, or at least to non-universal corrections to the conventional asymptotic behaviour. Some implications for the thermodynamic properties of random magnetic chains are also discussed.

Localization length of one-dimensional random systems

Considering the behavior at infinity of the amplitude of wave functions in one-dimensional disordered systems, as effectively described by a wave transmission through an infinitely large cluster, we derive an exact expression for the decay length ${L}_{d}(E)$ of eigenstates around energy $E$. We obtain from it simple formulas for ${L}_{d}$ at $E=0$ and in the limit of weak and strong disorder, and verify them in the case of Lorentzian type of disorder where exact results are independently available. The empirical result of Bush concerning the behavior of the maximum ${L}_{d}$ in random binary alloys is found to be approximately correct, but only in the limit of weak disorder.

Magnetic Properties of the One-Dimensional Random Ising System of Higher Spin

Magnetic Properties of the One-Dimensional Random Ising System of Higher Spin

Degree of Localization for the Eigenstates in One-Dimensional Random Systems

By solving integral equations, approximate stationary probability densities of some random variables associated with two kinds of random processes are obtained. The degrees of localization L(E) and L((J)) are calculated therefrom analytically over almost whole range of energy or frequency for several models of the one-dimensional random system. It is found that L(E) and L((J)) are always positive as was proved most generally by Matsuda and Ishii and that they are proportional to the variance of the . random variables characterizing the random system.

Magnetic Properties of One-Dimensional RandomXYModel

The magnetic properties of random magnetic systems have been a subject of interest in past years. Katsura and Tsujiyama1> studied one-dimensional random Ising model of S=1/2 with nearest-neighbor interaction and obtained the free energy and the susceptibility at zero magnetic field. Recently it is shown that the free energies of one-dimensional random magnetic systems with nearest-neighbor interaction can be given in terms of a power series of concentration of magnetic atoms.2> The free energy per one lattice site is

Localization in One-Dimensional Disordered Systems

The basic idea that the convergence of the renormalized perturbation expression for the self-energy ${\ensuremath{\Delta}}_{0}$ at a given energy is equivalent to the localizability of the eigenstates, if any, at this energy is applied to one-dimensional random systems, namely, electrons in the tightbinding approximation and phonons. For nearest-neighbor interactions all eigenstates are localized. If second-nearest-neighbor interactions are present, the possibility of the existence of extended states remains; we have shown that existing theories are unable to give a definite answer to the problem in this case.