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

Abstract

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. CONTENTS

Gap sets for the spectra of cubic graphs

“…The last gives a spectral characterization of the image of 𝒯; note that according to (3), −2 ∈ 𝜎(𝑌 ) for such 𝑌 's. Relation (iv) is well known (see [18]), while (v) was derived and exploited in our recent paper [21] and makes use of the characterization of graphs whose spectra are contained in [−2, ∞) ("Hoffman" graphs) [5]. 𝒯 provides us with a versatile tool to construct spectral sets.…”

“…In combinatorics and engineering applications, a gap at 3 defines "cubic expanders", an apparently very fruitful structure [19]. In our recent work [21] on microwave coplanar waveguide resonators it is the gap at the bottom −3 that is critical. In chemistry the stability properties of carbon fullerene molecules are dictated by the gap at 0 for the case of closed shells [11].…”

Hyperbolic band theory

“…Spurred by this experimental breakthrough, recent theoretical studies have explored the propagation of matter waves on hyperbolic lattices. Using graph theory and numerical diagonalization, Kollár et al (10) obtained general mathematical results concerning the existence of extended degeneracies and gaps in the spectrum of tight-binding Hamiltonians on a variety of discrete hyperbolic lattices. Boettcher et al (11) developed a hyperbolic analog of the effective-mass approximation in solid-state physics, showing that such tight-binding Hamiltonians reduce in the long-distance limit to the hyperbolic Laplacian-the Laplace-Beltrami operator associated with the Poincaré metric on the hyperbolic plane-and proposing the synthetic structures of Kollár et al (7) as a new platform for the simulation of quantum field theory in curved space.…”

Free Fermions Behind the Disguise

“…the edges in R are incident at a vertex in V . We note that line graphs also play an important role in understanding the spectrum of certain tightbinding models [56,57], but we will not discuss these models further here.…”