Level-spacing statistics for the Anderson model in one and two dimensions

Gap Statistics for Confined Particles with Power-Law Interactions

We consider the N particle classical Riesz gas confined in a one-dimensional external harmonic potential with power-law interaction of the form 1/r^{k}, where r is the separation between particles. As special limits it contains several systems such as Dyson's log-gas (k→0^{+}), the Calogero-Moser model (k=2), the 1D one-component plasma (k=-1), and the hard-rod gas (k→∞). Despite its growing importance, only large-N field theory and average density profile are known for general k. In this Letter, we study the fluctuations in the system by looking at the statistics of the gap between successive particles. This quantity is analogous to the well-known level-spacing statistics which is ubiquitous in several branches of physics. We show that the variance goes as N^{-b_{k}} and we find the k dependence of b_{k} via direct Monte Carlo simulations. We provide supporting arguments based on microscopic Hessian calculation and a quadratic field theory approach. We compute the gap distribution and study its system size scaling. Except in the range -1<k<0, we find scaling for all k>-2 with both Gaussian and non-Gaussian scaling forms.

A nanophotonic laser on a graph

Conventional nanophotonic schemes minimise multiple scattering to realise a miniaturised version of beam-splitters, interferometers and optical cavities for light propagation and lasing. Here instead, we introduce a nanophotonic network built from multiple paths and interference, to control and enhance light-matter interaction via light localisation. The network is built from a mesh of subwavelength waveguides, and can sustain localised modes and mirror-less light trapping stemming from interference over hundreds of nodes. With optical gain, these modes can easily lase, reaching ~100 pm linewidths. We introduce a graph solution to the Maxwell's equation which describes light on the network, and predicts lasing action. In this framework, the network optical modes can be designed via the network connectivity and topology, and lasing can be tailored and enhanced by the network shape. Nanophotonic networks pave the way for new laser device architectures, which can be used for sensitive biosensing and on-chip optical information processing.

From closed to open one-dimensional Anderson model: transport versus spectral statistics

Using the phenomenological expression for the level spacing distribution with only one parameter 0 ≤ β ≤ ∞ covering all regimes of chaos and complexity in a quantum system, we show that transport properties of the one-dimensional Anderson model of finite size can be expressed in terms of this parameter. Specifically, we demonstrate a strictly linear relation between β and the normalized localization length for the whole transition from strongly localized to extended states. This result allows one to describe all transport properties in the open system entirely in terms of the parameter β and the strength of the coupling to the continuum. For nonperfect coupling, our data show a quite unusual interplay between the degree of internal chaos defined by β and the degree of openness of the model. The results can be experimentally tested in single-mode waveguides with either bulk or surface disorder.