Self-Consistent Theory of Anderson Localization: General Formalism and Applications

The self-consistent theory of Anderson localization of quantum particles or classical waves in disordered media is reviewed. After presenting the basic concepts of the theory of Anderson localization in the case of electrons in disordered solids, the regimes of weak and strong localization are discussed. Then the scaling theory of the Anderson localization transition is reviewed. The renormalization group theory is introduced and results and consequences are presented. It is shown how scale-dependent terms in the renormalized perturbation theory of the inverse diffusion coefficient lead in a natural way to a self-consistent equation for the diffusion coefficient. The latter accounts quantitatively for the static and dynamic transport properties except for a region near the critical point. Several recent applications and extensions of the self-consistent theory, in particular for classical waves, are discussed.Comment: 25 pages, 2 figures; published version including correction

Noncommutative localization in algebraic LL-theory

Given a noncommutative (Cohn) localization $A \to \sigma^{-1}A$ which is injective and stably flat we obtain a lifting theorem for induced f.g. projective $\sigma^{-1}A$-module chain complexes and localization exact sequences in algebraic $L$-theory, matching the algebraic $K$-theory localization exact sequence of Neeman and Ranicki.Comment: to appear in Advances in Mathematic

Algebraic K-theory of strict ring spectra

We view strict ring spectra as generalized rings. The study of their algebraic K-theory is motivated by its applications to the automorphism groups of compact manifolds. Partial calculations of algebraic K-theory for the sphere spectrum are available at regular primes, but we seek more conceptual answers in terms of localization and descent properties. Calculations for ring spectra related to topological K-theory suggest the existence of a motivic cohomology theory for strictly commutative ring spectra, and we present evidence for arithmetic duality in this theory. To tie motivic cohomology to Galois cohomology we wish to spectrally realize ramified extensions, which is only possible after mild forms of localization. One such mild localization is provided by the theory of logarithmic ring spectra, and we outline recent developments in this area.Comment: Contribution to the proceedings of the ICM 2014 in Seou

Weak localization of Dirac fermions in graphene beyond the diffusion regime

We develop a microscopic theory of the weak localization of two-dimensional massless Dirac fermions which is valid in the whole range of classically weak magnetic fields. The theory is applied to calculate magnetoresistance caused by the weak localization in graphene and conducting surfaces of bulk topological insulators.Comment: 5 pages, 2 figure

Is Quantum Field Theory ontologically interpretable? On localization, particles and fields in relativistic Quantum Theory

In this paper, I provide a formal set of assumptions and give a natural criterion for a quantum field theory to admit particles. I construct a na\"ive approach to localization for a free bosonic quantum field theory and show how this localization scheme, as a consequence of the Reeh-Schlieder theorem, fails to satisfy this criterion. I then examine the Newton-Wigner concept of localization and show that it fails to obey strong microcausality and thus is subject to a more general version of the Reeh-Schlieder theorem. I review approaches to quantum field theoretic explanations of particle detection events and explain how particles can be regarded as emergent phenomena of a relativistic field theory. In particular, I show that effective localization of Hilbert space vectors is equivalent to an approximate locality of observable algebras.Comment: 33 page

Analytical and numerical study of uncorrelated disorder on a honeycomb lattice

We consider a tight-binding model on the regular honeycomb lattice with uncorrelated on-site disorder. We use two independent methods (recursive Green's function and self-consistent Born approximation) to extract the scattering mean free path, the scattering mean free time, the density of states and the localization length as a function of the disorder strength. The two methods give excellent quantitative agreement for these single-particle properties. Furthermore, a finite-size scaling analysis reveals that all localization lengths for different lattice sizes and different energies (including the energy at the Dirac points) collapse onto a single curve, in agreement with the one-parameter scaling theory of localization. The predictions of the self-consistent theory of localization however fail to quantitatively reproduce these numerically-extracted localization lengths.Comment: 19 pages, 25 figure