Non-Commutative Tools for Topological Insulators

This paper reviews several analytic tools for the field of topological insulators, developed with the aid of non-commutative calculus and geometry. The set of tools includes bulk topological invariants defined directly in the thermodynamic limit and in the presence of disorder, whose robustness is shown to have non-trivial physical consequences for the bulk states. The set of tools also includes a general relation between the current of an observable and its edge index, relation that can be used to investigate the robustness of the edge states against disorder. The paper focuses on the motivations behind creating such tools and on how to use them.Comment: Final version (some arguments were corrected

Spontaneous Transitions in Quantum Mechanics

The problem of spontaneous pair creation in static external fields is reconsidered. A weak version of the conjecture proposed by G Nenciu (1980) is seated and proved. The method reduces the proof of the general conjecture to the study of the evolution, associated with a time dependent Hamiltonian, of a vector which is eigenvector of this Hamiltonian at some given time. Possible ways of proving the general conjecture are discussed.Comment: An error was found and eliminate

Gaussian, Mean Field and Variational Approximation: the Equivalence

We show the equivalence between the three approximation schemes for self-interacting (1+1)-D scalar field theories. Based on rigorous results of [1, 2], we are able to prove that the Gaussian approximation is very precise for certain limits of coupling constants. The $\lambda \phi ^{4}+\sigma \phi ^{2}$ model will be used as a concrete application.Comment: 18 pages, no figur

The dielectric behavior of the living cell suspensions

In the limit of small concentrations and weak applied electric fields, the dielectric permittivity of suspensions of arbitrarily shaped, shelled and charged particles is calculated. It is proved that the dielectric behavior at low frequencies is dominated by the effects of the diffusion of the free charges on the shell surfaces. Our theoretical formula is valid in the low range of frequencies (alpha dispersion) as well as in the high range of frequencies (beta dispersion). Will result that one can measure the membrane electrical potential by a simple investigation of the living cell suspension dielectric properties.Comment: 16 pages, 13 figure

Transport properties of fermionic systems

We extend the method discovered by A Y Alekseev et al to the case of fermions in external fields. A general formula for conductance G is proved. In the (1+1)-D case with symmetry at time reflection, it is shown that: G=e^2/h+o(a^), where a is the strength of the external field. In (3+1)-D free case, it is checked that G=n*e^2/h, where n is the number of the filled energetic bands of the transversal quantization.Comment: 16 pages, no figure, new style + minor correction

On the Kohn-Sham equations with periodic background potentials

We study the question of existence and uniqueness for the finite temperature Kohn-Sham equations. For finite volumes, a unique soluion is shown to exists if the effective potential satisfies a set of general conditions and the coupling constant is smaller than a certain value. For periodic background potentials, this value is proven to be volume independent. In this case, the finite volume solutions are shown to converge as the thermodynamic limit is considered. The local density approximation is shown to satisfy the general conditions mentioned above

Electronic structure and optical properties of metallic nanoshells

The electronic structure and optical properties of metallic nanoshells are investigated using a jellium model and the Time Dependent Local Density Approximation (TDLDA). An efficient numerical implementation enables applications to nanoshells of realistic size with up to a million electrons. We demonstrate how a frequency dependent background polarizability of the jellium shell can be included in the TDLDA formalism. The energies of the plasmon resonances are calculated for nanoshells of different sizes and with different dielectric cores, dielectric embedding media, and dielectric shell backgrounds. The plasmon energies are found to be in good agreement with the results from classical Mie scattering theory using a Drude dielectric function. A comparison with experimental data shows excellent agreement between theory and the measured frequency dependent absorption spectra

Band Alignment in Molecular Devices: Influence of Anchoring Group and Metal Work Function

We present periodic Density Functional Theory calculations of the electronic properties of molecular junctions formed by amine-, and thiol-terminated alkane chains attached to two metal (Au, Ag) electrodes. Based on extensive analysis that includes molecular monolayers of varying densities, we establish a relationship between the alignment of the molecular energy levels and the interface dipoles, which shows that the band alignment (BA) in the limit of long, isolated chains is independent of the link group and can be computed from a reference system of non interacting molecule + metal electrodes. The main difference between the amine and thiol linkers is the effective dipole moment at the contact. This is very large, about 4.5 D, for amine linkers, leading to a strong dependence of the BA on the monolayer density and a slow convergence to the isolated molecule limit. Instead, this convergence is fast for S anchors due to the very small, ~ 0.2 D, effective dipoles at the contacts

Transport properties of the fermionic systems

We extend the method discovered by A Y Alekseev et al to the case of fermions in external fields. A general formula for conductance G is proved. It is shown that in (1+1)-D case: G=e^2/h+o(a^2), where a is the strength of the external field. In (3+1)-D free case, it is checked that G=n*e^2/h, where n is the number of the filled energetic bands of the transversal quantization

Transfer matrices for scalar fields on curved spaces

We apply Nelson's technique of constructing Euclidean fields to the case of classical scalar fields on curved spaces. It is shown how to construct a transfer matrix and, for a class of metrics, the basic spectral properties of its generator are investigated. An application concerning decoupling of non-convex disjoint region is given.Comment: 11 pages, no figur