Edge solitons in a nonlinear mechanical topological insulator

Abstract

We report localized and unidirectional nonlinear traveling edge waves discovered theoretically and numerically in a 2D mechanical (phononic) topological insulator. The lattice consists of a collection of pendula with weak Duffing nonlinearity connected by linear springs. We show that the classical 1D nonlinear Schrödinger equation governs the envelope of 2D edge modes, and study the propagation of traveling waves and rogue waves in 1D as edge solitons in 2D. As a result of topological protection, these edge solitons persist over long time intervals and through irregular boundaries.Topological insulators (TIs) are a unique state of matter with behavior that has significant ramifications in the fields of both condensed matter physics and optics. The importance of TIs has been established in the condensed matter community for almost a decade, with extensive literature on both one-dimensional and multidimensional systems [1][2][3]. The one stand out property of TIs which holds the ongoing interest in the topic is that a clear dichotomy exists between the edge (surface) and the bulk of the material as electrons are conducted only on the edge whilst the bulk is insulating. The existence of such edge states at the interface between two bulk materials with different topological invariants is guaranteed by the principle of bulk-edge correspondence [4,5]. More recently, the theoretical framework underlying quantum TIs has been generalized to photonic systems governed by classical electromagnetic fields [6][7][8]. In analogy to electrons in traditional TIs, electromagnetic waves in photonic TIs propagate along the edge with very little backscattering, even in the presence of disorders such as missing site(s) on the edge of a photonic lattice [9][10][11].The emerging field of topological mechanics utilizes such topological principles to reveal new collective excitations in classical mechanical (phononic) systems [12,13]. These topological acoustic metamaterials can be classified into two families depending on whether the topological edge modes appear at zero frequency or high frequencies. In the zero frequency case, these edge modes are identified as floppy modes and self-stress states in Maxwell frames [14]. Here we focus on the high frequency case, where topologically protected transport via phonons is enabled. Seminal work in this direction includes analogues of the quantum Hall effect using a lattice of hanging gyroscopes [15], and the quantum spin Hall effect (QSHE) using a lattice of coupled pendula [16] and bi-layered lattices of disks and springs [17].Over the last century, the theory of nonlinear waves in continuous systems has formed a cornerstone of nonlinear science [18]. A fundamental result is that in such systems, dispersion can balance with nonlinearity to produce robust localized nonlinear traveling waves known as solitons. The theory of nonlinear waves in discrete systems has flourished more recently, especially in the context of nonlinear optics and Bose-Einstein condensates [19]. In mechanical lattices, t...

Topological wave insulators: a review

“…For instance, in order to realize spin Hall phases in the acoustic lattice crystal discussed in Figure 1g and h, the extra cell coupling between resonators are necessarily required to be larger than the intra-cell ones. Amplitude-dependent nonlinear phenomena [347][348][349][350][351][352], however, can be utilized to achieve the same kind of topological phases without this special condition. In [353], Hadad et al investigated a one-dimensional nonlinear topological system based on a SSH toy model with nonlinear staggered potentials (Figure 4a).…”

Nonlinearity and Topology

“…There have been numerous recent efforts in this direction of harnessing topological properties of suitable mechanical media to improve the propagation or storage of energy. These include, among others, the examination of edge solitons [171] and their ability for nonlinear conduction in topological mechanical insulators [172,173], the study of nonlinear edge states that arise in phononic lattices [174], as well as the examination of topological band transitions in tunable phononic systems, under the variation of suitable (e.g. stiffness) parameters [175,176].…”

Topological Phononics: From Fundamental Models to Real Materials

“…Recently, there is growing research of interest in exploring topological states of phonons. Various topological phononic states have been proposed/realized in acoustic systems, mechanical metamaterials, Maxwell frames, and solid‐state materials . Topological phononic states are characterized by nonzero topological invariants.…”