Topological Dimensions from Disorder and Quantum Mechanics?

We have recently shown that the critical Anderson electron in D=3 dimensions effectively occupies a spatial region of the infrared (IR) scaling dimension dIR≈8/3. Here, we inquire about the dimensional substructure involved. We partition space into regions of equal quantum occurrence probabilities, such that the points comprising a region are of similar relevance, and calculate the IR scaling dimension of each.

Counting-Based Effective Dimension and Discrete Regularizations.

Fractal-like structures of varying complexity are common in nature, and measure-based dimensions (Minkowski, Hausdorff) supply their basic geometric characterization. However, at the level of fundamental dynamics, which is quantum, structure does not enter via geometric features of fixed sets but is encoded in probability distributions on associated spaces. The question then arises whether a robust notion of the fractal measure-based dimension exists for structures represented in this way.

Super-Universality in Anderson Localization.

We calculate the effective spatial dimension d_{IR} of electron modes at critical points of 3D Anderson models in various universality classes (O,U,S,AIII). The results are equal within errors, and suggest the super-universal value d_{IR}=2.665(3)≈8/3.

Transmission properties and effective electromagnetic parameters of double negative metamaterials.

We analyze the transmission properties of double negative metamaterials (DNM). Numerical simulations, based on the transfer matrix algorithm, show that some portion of the electromagnetic wave changes its polarization inside the DNM structure. As the transmission properties depend strongly on the polarization, this complicates the interpretation of experimental and numerical data, both inside and outside of the pass band.