Abstract:
We derive and discuss a technique for manipulating power series which is complementary to standard procedures. We begin with the translation operator, but we express the operator as an infinite product instead of expanding it as a series and we apply combinatorial arguments to generate the terms in the series in an efficient manner with a minimum of clutter and intermediate calculations. The method is effective for developing multivariate expansions, and may also be used to manipulate series, e.g. in operations where one must take the reciprocal of a power series or raise it to a power that may be fractional or irrational. In the case of two component perturbations, we obtain analytic expressions for the expansion coefficients. We use our technique to generate an electrostatic multipole expansion as a demonstration of its utility in producing coefficients of special functions such as the Legendre and Hermite polynomials.

Abstract:
In a tight binding framework, we analyze the characteristics of electronic states in strongly disordered materials (hopping sites are placed randomly with no local order) with tunneling matrix elements decaying exponentially in the atomic separation with various decay ranges l examined. We calculate the density of states (DOS) and the Inverse Participation Ratio (IPR) for amorphous atomic configurations in one, two, and three dimensions. With a finite size scaling analysis of the IPR statistical distributions, it is shown that states are either extended or localized for a particular energy, and phase portraits for wave functions are obtained showing extended and localized behavior in the thermodynamic limit. While we conclude that all states are localized in 1D, in the 2D case there is a threshold for l above which some eigenstates appear to be extended and below which wave functions are entirely localized. For 3D geometries, there are two mobility boundaries flanking an intermediate range of energies where states are extended with eigenstates localized for energies above or below this range. While a zone of extended states persists even for very short l, the width of the region tends to zero exponentially (i.e. scaling as exp{-A/l}) for very small decay length scales.

Abstract:
We calculate mean square deviations for crystals in one and two dimensions. For the two dimensional lattices, we consider several distinct geometries (i.e. square, triangular, and honeycomb), and we find the same essential phenomena for each lattice structure. We investigate the stability of long-range crystalline order for a variety of coupling schemes, including short-range exponentially decaying inter-atomic potentials and long-range interactions with a power law dependence r^{-alpha}. For the latter in the 1D case, we find a critical value alpha_c(1D) = 1.615 +/- 0.005 for the power law decay exponent below which crystalline order is intact, and above which thermal fluctuations destroy long-range order when T > 0. The corresponding critical value for two dimensional lattices with displacements confined to the plane is alpha_c(2D) = 3.15 +/ 0.025. If motion perpendicular to the crystal plane is permitted, thermally induced distortions diverge rapidly (i.e. linearly) in dual layer systems with local stiffness provided by an extended coupling scheme, even if the interaction is long ranged, decaying as a power law in the separation between lattice sites.

Abstract:
We calculate the phonon density of states (DOS) for strongly amorphous materials with a short-ranged interatomic potential. Exponentially decaying and abruptly truncated interatomic potentials are examined. Thermally excited mean square deviations from equilibrium are calculated with rapid increases noted as the average number of neighbors is reduced. The Inverse Participation Ratio (IPR) is used to characterize the phonon states and identify localized phonon modes as the bonding range (and hence the average number of neighbors per atom) is diminished. For the truncated potential, the characteristics of the IPR histogram change qualitatively below $n_{\mathrm{neigh}}$ with the appearance of localized phonon modes below $n_{\mathrm{neigh}} = 6.0$.

Abstract:
With a tight binding treatment we examine amorphous conductors with gas-like disorder, or no correlations among the site positions. We consider an exponentially decaying hopping integral with range $l$, and the Inverse Participation Ratio (IPR) is used to characterize carrier wave functions with respect to localization. With the aid of two complementary finite size scaling techniques to extrapolate to the bulk limit (both methods exploit critical behavior in different ways to find the boundary between domains of extended and localized wave functions) which nevertheless yield identical results, we obtain phase diagrams showing regions where states are extended and domains of localized states. In the 2D case, states are localized below a threshold length scale $l_{c}$ on the order of the interparticle separation $\rho^{-1/2}$ with a finite fraction of states extended for $l > l_{c}$. For $D = 3$, the extended phase is flanked by regions of localized states and bounded by two mobility edges. The swath of extended states, broad for $l \sim 1$, becomes narrower with decreasing $l$, though persisting with finite width even for $l < (1/5)\rho^{-1/3}$. Mobility edges are interpreted as lines of critical points, and we calculate the corresponding critical exponents.

Abstract:
The percolation threshold for flow or conduction through voids surrounding randomly placed spheres is rigorously calculated. With large scale Monte Carlo simulations, we give a rigorous continuum treatment to the geometry of the impenetrable spheres and the spaces between them. To properly exploit finite size scaling, we examine multiple systems of differing sizes, with suitable averaging over disorder, and extrapolate to the thermodynamic limit. An order parameter based on the statistical sampling of stochastically driven dynamical excursions and amenable to finite size scaling analysis is defined, calculated for various system sizes, and used to determine the critical volume fraction phi_{c} = 0.0317 +/- 0.0004 and the correlation length exponent nu = 0.92 +/- 0.05.

Abstract:
We calculate Root Mean Square (RMS) deviations from equilibrium for atoms in a two dimensional crystal with local (e.g. covalent) bonding between close neighbors. Large scale Monte Carlo calculations are in good agreement with analytical results obtained in the harmonic approximation. When motion is restricted to the plane, we find a slow (logarithmic) increase in fluctuations of the atoms about their equilibrium positions as the crystals are made larger and larger. We take into account fluctuations perpendicular to the lattice plane, manifest as undulating ripples, by examining dual layer systems with coupling between the layers to impart local rigidly (i.e. as in sheets of graphene made stiff by their finite thickness). Surprisingly, we find a rapid divergence with increasing system size in the vertical mean square deviations, independent of the strength of the interplanar coupling. We consider an attractive coupling to a flat substrate, finding that even a weak attraction significantly limits the amplitude and average wavelength of the ripples. We verify our results are generic by examining a variety of distinct geometries, obtaining the same phenomena in each case.

Abstract:
Using large-scale Monte Carlo calculations, we consider strongly disordered Heisenberg models on a cubic lattice with missing sites (as in diluted magnetic semiconductors such as Ga_{1-x}Mn_{x}As). For disorder ranging from weak to strong levels of dilution, we identify Curie temperatures and calculate the critical exponents nu, gamma, eta, and beta finding, per the Harris criterion, good agreement with critical indices for the pure Heisenberg model where there is no disorder component. Moreover, we find that thermodynamic quantities (e.g. the second moment of the magnetization per spin) self average at the ferromagnetic transition temperature with relative fluctuations tending to zero with increasing system size. We directly calculate effective critical exponents for T > T_{c}, yielding values which may differ significantly from the critical indices for the pure system, especially in the presence of strong disorder. Ultimately, the difference is only apparent, and eventually disappears when T is very close to T_{c}.

Abstract:
We consider ferromagnetism in spatially randomly located magnetic moments, as in a diluted magnetic semiconductor, coupled via the carrier-mediated indirect exchange RKKY interaction. We obtain via Monte Carlo the magnetic phase diagram as a function of the impurity moment density $n_{i}$ and the relative carrier concentration $n_{c}/n_{i}$. As evidenced by the diverging correlation length and magnetic susceptibility, the boundary between ferromagnetic (FM) and non-ferromagnetic (NF) phases constitutes a line of zero temperature critical points which can be viewed as a magnetic percolation transition. In the dilute limit, we find that bulk ferromagnetism vanishes for $n_{c}/n_{i}>.1$. We also incorporate the local antiferromagnetic direct superexchange interaction between nearest neighbor impurities, and examine the impact of a damping factor in the RKKY range function.

Abstract:
Using Ginzburg-Landau theory, we have performed detailed studies of vortices in the presence of artificial defect arrays, for a thin film geometry. We show that when a vortex approaches the vicinity of a defect, an abrupt transition occurs in which the vortex core develops a ``string'' extending to the defect boundary, while simultaneously the supercurrents and associated magnetic flux spread out and engulf the defect. Current induced depinning of vortices is shown to be dominated by the core string distortion in typical experimental situations. Experimental consequences of this unusual depinning behavior are discussed.