Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Electrodynamics

Materials science and the study of the electronic properties of solids are a major field of interest in both physics and engineering. The starting point for all such calculations is single-electron, or non-interacting, band structure calculations, and in the limit of strong on-site confinement this can be reduced to graph-like tight-binding models. In this context, both mathematicians and physicists have developed largely independent methods for solving these models. In this paper we will combine and present results from both fields. In particular, we will discuss a class of lattices which can be realized as line graphs of other lattices, both in Euclidean and hyperbolic space. These lattices display highly unusual features including flat bands and localized eigenstates of compact support. We will use the methods of both fields to show how these properties arise and systems for classifying the phenomenology of these lattices, as well as criteria for maximizing the gaps. Furthermore, we will present a particular hardware implementation using superconducting coplanar waveguide resonators that can realize a wide variety of these lattices in both non-interacting and interacting form

Quantum Graphs II: Some spectral properties of quantum and combinatorial graphs

The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics, and other areas. A Schnol type theorem is proven that allows one to detect that a point belongs to the spectrum when a generalized eigenfunction with an subexponential growth integral estimate is available. A theorem on spectral gap opening for ``decorated'' quantum graphs is established (its analog is known for the combinatorial case). It is also shown that if a periodic combinatorial or quantum graph has a point spectrum, it is generated by compactly supported eigenfunctions (``scars'').Comment: 4 eps figures, LATEX file, 21 pages Revised form: a cut-and-paste blooper fixe

Pseudo-random graphs

Random graphs have proven to be one of the most important and fruitful concepts in modern Combinatorics and Theoretical Computer Science. Besides being a fascinating study subject for their own sake, they serve as essential instruments in proving an enormous number of combinatorial statements, making their role quite hard to overestimate. Their tremendous success serves as a natural motivation for the following very general and deep informal questions: what are the essential properties of random graphs? How can one tell when a given graph behaves like a random graph? How to create deterministically graphs that look random-like? This leads us to a concept of pseudo-random graphs and the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page

Effect of polycrystallinity on the optical properties of highly oriented ZnO grown by pulsed laser deposition

We report the results of photoluminescence and reflectance measurements on highly c-axis oriented polycrystalline ZnO grown by pulsed laser deposition. The samples measured were grown under identical conditions and were annealed in-situ at various temperatures for 10-15 min. The band-edge photoluminescence spectra of the material altered considerably with an increase in grain size, with increased free exciton emission and observable excitonic structure in the reflectance spectra. The green band emission also increased with increasing grain size. A deformation potential analysis of the effect of strain on the exciton energy positions of the A- and B-excitons demonstrated that the experimental exciton energies could not be explained solely in terms of sample strain. We propose that electric fields in the samples due to charge trapping at grain boundaries are responsible for the additional perturbation of the excitons. This interpretation is supported by theoretical estimates of the exciton energy perturbation due to electric fields. The behaviour of the green band in the samples provides additional evidence in favour of our model