Partition function for an electron in a random potential

We compute the average partition function for an electron moving in a Gaussian random potential. A path integral formulation is used, with a trial action like that in Feynman's polaron theory. We compute the variational bound as well as the first correction in a systematic cumulant expansion. The results are checked against exact formulas for the onedimensional white noise problem. The density of states in the low-energy tail has the correct exponential energy dependence, and energy-dependent prefactor to within a few percent. In addition, the partition function goes over smoothly to the perturbation theory result at high temperatures.Work supported by the National Science Foundation.     

Influence of a quasiclassical random potential on electron correlations in a one-dimensional system

The influence of a quasiclassical random potential on the behaviour of the correlation functions in a one-dimensional electron system is investigated in the generalized Luttinger model. We prove rigorously that in the case of a not-half-filled band the quasiclassical electron scattering suppresses dielectric and antiferromagnetic correlations, but has no effect on superconductive correlations. The case of a half-filled band, in which the existence of a gap Δ_ρ in the density excitation spectrum results in the exponential decrease of superconductive correlations in the pure system, is also considered. It is shown that in the “dirty” limit, Δ_ρτ ⪡ 1 (τ is the electron relaxation time), the quasiclassical random potential recovers a long-range, power law behaviour of superconductive correlations, typical for a gapless regime.     

Excitations of one-dimensional Bose-Einstein condensates in a random potential

We examine bosons hopping on a one-dimensional lattice in the presence of a random potential at zero temperature. Bogoliubov excitations of the Bose-Einstein condensate formed under such conditions are localized, with the localization length diverging at low frequency as l(omega) approximately 1/omega(alpha). We show that the well-known result alpha=2 applies only for sufficiently weak random potential. As the random potential is increased beyond a certain strength, alpha starts decreasing. At a critical strength of the potential, when the system of bosons is at the transition from a superfluid to an insulator, alpha=1. This result is relevant for understanding the behavior of the atomic Bose-Einstein condensates in the presence of random potential, and of the disordered Josephson junction arrays.     

Statics and dynamics of 2D colloidal crystals in a random pinning potential

We report the first experimental study of a model system of a two-dimensional colloidal crystal in a random pinning potential. The colloidal crystal consists of monodispersed charged polystyrene microspheres suspended in deionized aqueous media and confined near a rough charged surface. It is found that the static orientational correlation function g6(r) decays exponentially for intermediate and strong pinning, in agreement with theories. The driven depinning is dominated by thermally activated creep motion along 1D-like channels between regions with short-range order. A coexistence model is proposed for describing the observations.     

Drift,creep and pinning of a particle in a correlated random potential

The motion of a particle in a correlated random potential under the influence of a driving force is investigated in mean field theory. The correlations of the disorder are characterized by a short distance cutoff and a power law decay with exponent γ at large distances. Depending on temperature and γ, drift with finite mobility, creep or pinning is found. This is in qualitative agreement with results in one dimension. This model is of interest not only in view of the motion of particles or manifolds in random media, it also improves the understanding of glassy non-equilibrium dynamics in mean field models. The results, obtained by numerical integration and analytic investigations of the various scaling regimes in this problem, are compared with previous proposals regarding the long time properties of such systems and with replica calculations.     

Mobility of an electron in a Guassian random potential

The mobility of an electron in a Gaussian random potential is evaluated. It is shown that the relaxation time of the electron is found to be proportional to $\text{L}{}^2\text{|}\text{p}\text{|}$ for large and small correlation lengths L where |p| is the electron momentum.     

Diffusion with random traps: Transient one-dimensional kinetics in a linear potential

The problem of one-dimensional diffusion with random traps is solved without and with a constant field of force. Using an eigenvalue expansion for long times and the method of images for short times we give an exact, straightforward solution for the time dependence of the mean survival probability and the mean probability density for returning to the origin. Using the backward equation approach, we determine the mean survival time and the mean residence time density at the origin. We comment on the relation between these solutions and those for one-dimensional diffusion with random reflectors.     

一维抛物势场中单电子量子比特的性质

根据-维抛物势场中单电子能量本征波函数的特点,以基态和第二激发态波函数构造一个量子比特,并研究量子比特的性质.数值计算表明,量子比特内电子的概率密度随空间坐标和时间的变化而变化,在平衡位置概率密度幅值最大,在其他位置相对较小;且各个空间点的概率密度均随时间做周期性振荡,振荡周期完全由抛物势场的性质决定.     

Renormalization of potential scattering in a weakly interacting one-dimensional electron gas

The conductance of a weakly interacting electron gas in the presence of a single scatterer is found at arbitrary strength of the scattering potential. At weak interaction one can use a simple renormalization group approach instead of the standard bosonization technique. For a model with spinless electrons this approach allows us to show explicitly the crossover from the Fermi-gas to the low-temperature Luttinger liquid behavior. Deviations from the Luttinger liquid theory are studied for a realistic model of spin- electrons.     

Green's function for an electron in one-dimensional conductor with finite correlation length of impurity potential

The scattering of electrons by impurities in one-dimensional conductor is treated as a result of their interaction with quantized fields, the excitons of which carry the quanta of electronic relaxation time or mean free path. The relaxation process is represented as a consequences of exchange of those quanta between the electrons and is shown to be affected by fluctuations of dissipative fields. The Green's function of electrons is calculated and the dependence of the full mean free path on the electronic energy and on the scattering parameters is discussed.     

CDW instability of electron gas in a strong magnetic field

It is shown that within the Hartree-Fock approximation the electron gas in the quantum limit of a strong magnetic field has an instability as the temperature is lowered toward the charge density wave state with an wave-vector which has finite components in the directions not only parallel but also perpendicular to the field. The critical temperature, T_c, is estimated under the assumption of T_c??_F?ω_c, where ?_F and ω_c are the Fermi energy of the non-interacting system and the cyclotron frequency respectively.     

Ground-state and quenched-state properties of a one-dimensional interacting lattice gas in a random potential

We determine the zero-temperature properties of a one-dimensional lattice gas of particles that interact via a nearest neighbor exclusion potential and are subject to a random external field. The model is a special limiting case of the random field Ising chain. We calculate (1) the energy and density of the ground state as well as the local energy-density correlation and (2) the pair correlation function. The latter calculation gives access to all higher order correlations. The structure factor is shown to be a squared Lorentzian. We also compare the ground state to the quenched state obtained by sequentially filling the lowest available energy levels.     

Localization of random walks in one-dimensional random environments

We consider a random walk on the one-dimensional semi-lattice ={0, 1, 2,...}. We prove that the moving particle walks mainly in a finite neighbourhood of a point depending only on time and a realization of the random environment. The size of this neighbourhood is estimated. The limit parameters of the walks are also determined.     

Persistent random walks in a one-dimensional random environment

Central limit theorems are obtained for persistent random walks in a onedimensional random environment. They also imply the central limit theorem for the motion of a test particle in an infinite equilibrium system of point particles where the free motion of particles is combined with a random collision mechanism and the velocities can take on three possible values.Work supported by the Central Research Fund of the Hungarian Academy of Sciences (grant No. 476/82).     

Diffusion in random one-dimensional systems

Diffusion on the one-dimensional lattice is described by a master equation with nearest-neighbor transfer rates (symmetric or asymmetric). The transfer rates associated with bonds are assumed to be independent, equally distributed random variables. Under various conditions on their common distribution the large time behavior of averaged site probabilities and/or related quantities is exhibited.Presented at the Symposium on Random Walks, Gaithersburg, MD, June 1982.     

Loops in one-dimensional random walks

Distribution of loops in a one-dimensional random walk (RW), or, equivalently, neutral segments in a sequence of positive and negative charges is important for understanding the low energy states of randomly charged polymers. We investigate numerically and analytically loops in several types of RWs, including RWs with continuous step-length distribution. We show that for long walks the probability density of the longest loop becomes independent of the details of the walks and definition of the loops. We investigate crossovers and convergence of probability densities to the limiting behavior, and obtain some of the analytical properties of the universal probability density. Received 8 January 1999     

Vortex glass transition in a random pinning model

We study the vortex glass transition in disordered high temperature superconductors using Monte Carlo simulations. We use a random pinning model with strong point-correlated quenched disorder, a net applied magnetic field, long-range vortex interactions, and periodic boundary conditions. From a finite size scaling study of the helicity modulus, the rms current, and the resistivity, we obtain critical exponents at the phase transition. The new exponents differ substantially from those of the gauge glass model, but are close to those of the pure three-dimensional XY model.     

Tail estimates for one-dimensional random walk in random environment

Suppose that the integers are assigned i.i.d. random variables { _x } (taking values in the unit interval), which serve as an environment. This environment defines a random walk {X _k } (called a RWRE) which, when atx, moves one step to the right with probability _x , and one step to the left with probability 1- _x . Solomon (1975) determined the almost-sure asymptotic speed (=rate of escape) of a RWRE. For certain environment distributions where the drifts 2 _x -1 can take both positive and negative values, we show that the chance of the RWRE deviating below this speed has a polynomial rate of decay, and determine the exponent in this power law; for environments which allow only positive and zero drifts, we show that these large-deviation probabilities decay like exp(–Cn ^1/3). This differs sharply from the rates derived by Greven and den-Hollander (1994) for large deviation probabilities conditioned on the environment. As a by product we also provide precise tail and moment estimates for the total population size in a Branching Process with Random Environment.Partially supported by NSF DMS-9209712 and DMS-9403553 grants, by a US-ISRAEL BSF grant and by the S. and N. Grand research fund.Research partially supported by NSF grant # DMS-9404391 and a Junior Faculty Fellowship from the Regents of the University of California.Partially supported by NSF grant # DMS-9302709, by a US-Israel BSF grant and by the fund for promotion of research at the Technion.