Edge modes in modulated metamaterials based on the three-gap theorem

TL;DR

The paper addresses edge-localised states in metamaterials generated by the three-gap theorem, unifying SSH-like discretised couplings with Harper modulation. It develops a localization framework based on the factor $\alpha(\omega)$ and analyzes both finite and infinite (Floquet-Bloch) spectra, revealing how edge modes appear and disappear as the three-gap parameters vary. A key result is the link between finite-system edge states and the infinite-band structure, including a criterion for localisation and a transfer-matrix perspective on spectral flow. The findings enable design of complex edge-mode patterns with only three distinct couplings and provide a conceptual bridge between SSH-type topology and Harper modulation in modulated metamaterials.

Abstract

We present a new paradigm for generating complex structured materials based on the three-gap theorem that unifies and generalises several key concepts in the study of localised edge states. Our model has both the discretised coupling strengths of the SSH model and a modulation parameter that can be used to characterise the spectral flow of edge modes and produce images reminiscent of the Hofstadter butterfly. By defining a localisation factor associated to each eigenmode, we are able to establish conditions for the existence of localised edge states in finite systems. This allows us to compare their eigenfrequencies with the spectra of the corresponding infinitely periodic problem and characterise the rich pattern of localised edge modes appearing and disappearing (in the sense of becoming delocalised) as the parameters of our three-gap algorithm are varied.

Figures (4)

• Figure 1: Schematic of the three-gap algorithm for unit cell formation. Points on the circle are generated by repeatedly rotating a given angle $\theta$ and then uncurled onto a straight line. These points become the masses of a mass-spring system whose spring constants are inversely proportional to the distance between neighbouring points. On the right hand side, the spectral band structure corresponding to the infinitely periodic system is shown.
• Figure 2: Panel (a) is a phase space diagram showing eigenfrequencies $\omega^2$ vs. modulation parameter $\theta$ for the spectrum of the infinitely periodic system (black lines) and for the finite problem with $n=7$ unit cells (red lines). Blue dashed lines represent geometrical conditions for which the infinitely periodic and the finite spectrum share at least one of their eigenvalues. Panel (b) shows the evolution of the localisation factor $\alpha$ as a function of the modulation parameter for one of the eigenstates of the finite-sized problem that lies in the bandgap of the infinite system (the blue line indicates the critical value $|\alpha|=1$). Finally, panel (c) depicts four eigenstates corresponding to different geometrical structures that have been marked with colour markers in panels (a) and (b).
• Figure 3: Phase space diagrams showing eigenfrequencies $\omega^2$ vs. modulation parameter $\theta$ for different $N$ values. Black lines represent the spectra of the infinite systems, while red lines indicate the eigenfrequencies of the finite problem with $n = 5$ unit cells. Complexity of the space diagram quickly grows with increasing number of masses in the unit cell.
• Figure 4: Modulation angles at which at least one eigenmode of the single unit cell system $K$ has a localisation factor $|\alpha| = 1$ so is delocalised. Green circles show those configurations for which the resulting unit cell is made of all equal springs so is singly periodic. Golden diamonds represent configurations characterised in Proposition \ref{['prop:alpha1']}, with $\theta = (N-1)/2N$ for even $N$ or $\theta = (N-2)/2(N-1)$ for odd $N$ (in which case half of the eigenvalues are known to be delocalised). Blue squares show other geometrical configurations for which $|\alpha| = 1$ occurs.

Theorems & Definitions (18)

• Proposition 3.1
• proof
• Definition 3.1
• Proposition 3.2
• proof
• proof
• Remark
• Proposition 3.3
• proof
• proof