Probing the influence of topological and geometric disorder on the spectrum of the differential Laplacian operator on networks

TL;DR

This work generates compact metric network structures using the spatial tessellations of two-dimensional hyperuniform point patterns, which have suppressed large-scale density fluctuations relative to typical disordered point patterns, and characterize the eigenvalue spectrum structure of the differential Laplace operator on these networks.

Abstract

Metric networks are network-shaped, one-dimensional structures on which one can solve differential equations to simulate a wide range of physical systems including conjugated molecules, photonic crystals, quantum mechanics in waveguide networks, and acoustic metamaterials. More concretely, a metric network is a network whose edges are each assigned a notion of length and a coordinate describing position. One can then define function spaces and differential operators on these objects to model the aforementioned systems. Recent software advancements have made it feasible to analyze partial differential equations on large, compact metric networks with a vast array of structures. Here, we generate compact metric network structures using the spatial tessellations of two-dimensional hyperuniform point patterns, which have suppressed large-scale density fluctuations relative to typical disordered point patterns. This choice of structure is inspired by the exotic physical properties of network materials with these structures in other contexts. Then, we characterize the eigenvalue spectrum structure of the differential Laplace operator on these networks. In particular, we find that gaps can form in the eigenvalue spectra of these networks whose widths increase when the distribution of edge lengths is narrow and as the number of triangular faces increases. Importantly, many of the structures we consider are realizable in Euclidean space, meaning they are well-suited for practical applications in, e.g., metamaterial design. This work can thus be used to inform the design of metric network-based systems with spectral gaps with tunable widths and locations.

Figures (8)

• Figure 1: An example eigenmode on one of the disordered metric network structures considered herein. This network is generated from the Delaunay tessellation of a hyperuniform $A_2$ URL point pattern with $N = 418$ particles, $a = 0.1$, and prune boundary conditions (all of which are defined in Sec. \ref{['Sec:Methods']}). This eigenmode with eigenvalue $\lambda=-9.385$ corresponds to one in the "central island" described in Sec. \ref{['Sec:Results']}.
• Figure 1: Density $D(\lambda)$ of eigenvalues $\lambda$ for (a) standard and (b) equilateral metric networks generated from a Delaunay tessellation of a $\mathbb{Z}^2$ URL with $a=0.1$ with either clip, delete, or prune boundary conditions applied.
• Figure 2: Density $D(\lambda)$ of eigenvalues $\lambda$ for metric networks generated from a Delaunay-centroidal (C), Gabriel (G), or Delaunay (D) tessellation of (a) a $\mathbb{Z}^2$ URL with $a=0.1$ and (b) a disordered stealthy hyperuniform point pattern with $\chi = 0.48$, both with prune boundary conditions.
• Figure 3: Probability density of edge lengths $\ell_m$ in D networks generated from (a) $\mathbb{Z}^2$ URLs with $a = 0.05,0.1,0.2,$ and $0.3$ and (b) disordered stealthy hyperuniform point patterns with $\chi = 0.48$ and $0.40$, $A_2$ URLs with $a = 0.1$ and hard-disk fluids with $\phi = 0.65$.
• Figure 4: Density $D(\lambda)$ of eigenvalues $\lambda$ for metric networks generated from Delaunay tessellations of (a) $\mathbb{Z}^2$ URLs with $a = 0.05,0.1, 0.2$ and $0.3$ (b) disordered stealthy hyperuniform point patterns with $\chi = 0.48$ and $0.40$ and (c) an $A_2$ URL with $a = 0.1$, a disordered stealthy hyperuniform point pattern with $\chi = 0.48$, or a dark disk fluid point pattern with $\phi = 0.65$ with prune boundary conditions.