White Noise Potential Research Articles

Estimates for the state density for ordinary differential operators with white gaussian noise potential

SynopsisThe present paper is concerned with developing the existence and asymptotic properties of the state density N(λ) associated with certain higher order random ordinary differential operators A of the formwhere Ao has homogeneous and ergodic coefficients with respect to the σ-algebra generated by the Wiener process q(ω, x). The analysis uses the Weyl min-max principle to determine rough upper and lower bounds for N(λ).

New field theory formulation of localised states in disordered systems

The authors present a new field theoretic formulation of the quantum mechanics of disordered systems. The problem is converted to an explicit field theory by changing variables in the functional integral over all random potentials to an integral over all possible wave-functions. Unlike previous formulations this field theory has the novel feature of having the 'right' sign for the coefficient of the phi 4 term. Thus the important excitations are kinks rather than instantons. In addition, it exhibits singular solutions with finite action. The method is illustrated with an exact calculation of the asymptotic density of states in a one-dimensional gaussian white noise potential.

Disordered one-dimensional conductors at finite temperature

Abstract Inelastic scattering processes at nonzero temperatures produce a broadening, h/τin, of electronic energies which essentially is equivalent to the imaginary component of the energy one must introduce in order to evaluate the Kubo formula for the d.c. conductivity of a dirty metal. We1) have developed a fast algorithm for evaluating the conductivity of a linear chain, and have used it to study the dependence of the conductivity on the magnitude of the imaginary part of the electron energy. Results for the Anderson model with different degrees of disorder and at different energies can all be scaled onto the same curve, which is of the form expected from the usual theory of localized states. A cross-over is evident between two limiting regimes: in one the damping caused by the complex energy is dominant; in the other, localization of the eigenstates dominates. The universal curve obtained make it possible to connect tight binding model results with the conductivity calculated for an electron in a white noise potential. This curve describes models in which the scattering potentials have finite variance. Similar, but not identical, results are obtained for tight binding chains with a Cauchy distribution of site energies.

Conductivity of the disordered linear chain

The authors develop a fast algorithm for evaluating the Kubo formula for the conductivity of a linear chain, and use it to study the dependence of the conductivity as a function of imaginary frequency. The results for the Anderson model with different degrees of disorder and different energies can all be scaled onto the same curve, which is of the form expected from the theory of localised states. The universal curve obtained provides a simple connection between tight-binding model results and the conductivity which can be calculated for an electron in a white noise potential. Similar, but not identical, results are obtained for tight-binding chains with a Cauchy distribution of site energies.

On spectra of the Schr�dinger operator with a white Gaussian noise potential

On spectra of the Schr�dinger operator with a white Gaussian noise potential

Electrons in a four-dimensional disordered system

The problem of an electron moving in a random potential in four dimensions is examined. It is argued that the apparent simplicity of the white-noise potential is due to an improper limiting process. It is demonstrated that localised states occur when finite ranged potential fluctuations are considered, and that they do not disappear as the range of the fluctuations gets shorter. The Ioffe-Regel criterion for localisation is examined, and the coherent potential approximation is used to estimate the deviation of the density of states from a linear dependence on energy.

Phase transitions for the one-dimensional Landau-Ginzburg system and their relation to the theory of polymers and of electrons in a random potential

A study is made of the phase transitions for the one-dimensional Landau-Ginzburg system in the limit of zero degrees of freedom found by Balian and Toulouse (see abstr. A30922 of 1974). It is shown that they correspond to a rather simple property of a polymer in one dimension. The phase transitions are exhibited for a discrete model as well as for the Landau-Ginzburg model. The properties of an electron in a one-dimensional random potential can be found by an analytic continuation of this theory, but the phase transitions lie away from the physical region of this system. An explicit expression is derived for the localization length for the electron in a white-noise potential, and it is indicated how the conductivity could be calculated.

Impurity-Band Tails in the High-Density Limit. I. Minimum Counting Methods

A new approximate method is presented for calculating the density of states and one-electron Green's function in the low-energy tail of a high-density impurity band. Such states occur in regions of large attractive potential produced by unusually high (random) concentrations of attractive centers and/or unusually low concentrations of repulsive centers. These well-separated deep wells each have one bound state of lowest energy. The distribution of these lowest levels vields the low-energy tail. For any one well, a bound-state energy $E({\mathbf{x}}_{0})$ is estimated variationally using $\ensuremath{\psi}(\mathbf{x})=f(\mathbf{x}\ensuremath{-}{\mathbf{x}}_{0})$, where $f(\mathbf{x})$ has any fixed form. The best energy for any well is obtained by minimizing $E({\mathbf{x}}_{0})$ with respect to ${\mathbf{x}}_{0}$ in the vicinity of that well. The number of wells that contribute to the density of states at energy $E$ is given by the number of local minima in $E({\mathbf{x}}_{0})$ that occur at the level $E$. In the high-density limit, Gaussian statistics are adequate for treating the potential fluctuations and the expectation value for the density of states is easily calculated. At low energies the best choice for the function $f$ is that which maximizes the expected density of states. Application to an exactly soluble one-dimensional model, a Gaussian white noise potential, yields the correct asymptotic form $\ensuremath{\rho}(E)=\mathrm{const}|E|\mathrm{exp}[\ensuremath{-}(\frac{4}{3}){|2E|}^{\frac{3}{2}}]$. In three dimensions, the density of states in the Gaussian approximation is found to have the form $\ensuremath{\rho}(E)=[\frac{A(E)}{{\ensuremath{\xi}}^{2}}]\mathrm{exp}[\frac{\ensuremath{-}B(E)}{(2\ensuremath{\xi})}]$, where $\ensuremath{\xi}$ is proportional to the concentration of impurities and to the square of the strength of the impurity potential. For screened, charged impurities, $B(E)={E}_{{Q}^{2}}b(\ensuremath{\nu})$, $A(E)={Q}^{3}{E}_{{Q}^{3}}a(\ensuremath{\nu})$, where ${E}_{Q}=\frac{{\ensuremath{\hbar}}^{2}{Q}^{2}}{(2{m}^{*})}$, ($\frac{1}{Q}$) is the screening radius, and $\ensuremath{\nu}=\frac{({E}_{0}\ensuremath{-}E)}{{E}_{Q}}$ is the energy below the mean potential ${E}_{0}$ in units of ${E}_{Q}$. Computer-calculated curves are provided for the universal dimensionless functions $a(\ensuremath{\nu})$ and $b(\ensuremath{\nu})$. The exponent $b(\ensuremath{\nu})$ behaves like ${\ensuremath{\nu}}^{\frac{1}{2}}$ when $\ensuremath{\nu}$ is small (strong screening) and behaves like ${\ensuremath{\nu}}^{2}$ when $\ensuremath{\nu}$ is large (weak screening).