Topological dynamics of gyroscopic and Floquet lattices from Newton's laws

Abstract

Despite intense interest in realizing topological phases across a variety of electronic, photonic and mechanical platforms, the detailed microscopic origin of topological behavior often remains elusive. To bridge this conceptual gap, we show how hallmarks of topological modes -boundary localization and chirality -emerge from Newton's laws in mechanical topological systems. We first construct a gyroscopic lattice with analytically solvable edge modes, and show how the Lorentz and spring restoring forces conspire to support very robust "dangling bond" boundary modes. The chirality and locality of these modes intuitively emerges from microscopic balancing of restoring forces and cyclotron tendencies. Next, we introduce the highlight of this work, a very experimentally realistic mechanical non-equilibrium (Floquet) Chern lattice driven by AC electromagnets. Through appropriate synchronization of the AC driving protocol, the Floquet lattice is "pushed around" by a rotating potential analogous to an object washed ashore by water waves. Besides hosting "dangling bond" chiral modes analogous to the gyroscopic boundary modes, our Floquet Chern lattice also supports peculiar half-period chiral modes with no static analog. With key parameters controlled electronically, our setup has the advantage of being dynamically tunable for applications involving arbitrary Floquet modulations. The physical intuition gleaned from our two prototypical topological systems are applicable not just to arbitrarily complicated mechanical systems, but also photonic and electrical topological setups. arXiv:1701.03385v2 [cond-mat.mes-hall]

No-go theorem for topological insulators and high-throughput identification of Chern insulators

“…In particular, the applications extend to bosonic band systems such as photonic crystals, 101,102 phonon bands, 103,104 and linear circuit lattices. [66][67][68] Finally, we remark on our definition of a BR -as unitarily equivalent to locally-symmetric Wannier functions. A priori, this definition is not obviously equivalent to the conventional definition [49][50][51][52] of BRs as induced representations of space groups; this equivalence is proven in App.…”

Directional Control of Rayleigh Wave Propagation in an Elastic Lattice by Gyroscopic Effects

“…This type of metamaterial can be utilized to force waves to propagate along a line, whose direction is defined by the geometry of the medium (Carta et al, 2017). Gyroscopic spinners can also be employed to design topological insulators, where waves travel in one direction and are immune to backscattering (Nash et al, 2015;Süsstrunk and Huber, 2015;Wang et al, 2015;Garau et al, 2018;Lee et al, 2018;Mitchell et al, 2018;Carta et al, 2019;Garau et al, 2019;Carta et al, 2020). Furthermore, systems with gyroscopic spinners can be used to design coatings to hide the presence of objects in a continuous or discrete medium (Brun et al, 2012;Garau et al, 2019).…”

Rayleigh waves in micro-structured elastic systems: Non-reciprocity and energy symmetry breaking

“…The first model of a gyroscopic elastic lattice in which each nodal mass is connected to a gyroscopic spinner was proposed by Brun et al (2012). The gyroscopic effect can be employed to tune the frequency at which dynamic anisotropy is observed, as shown in the work of Carta et al (2014), to create extremely localised waveforms as demonstrated by Carta et al (2017), to force waves to propagate along a preferential direction illustrated in the works of Nash et al (2015), Süsstrunk & Huber (2015), Wang et al (2015), Garau et al (2018), Mitchell et al (2018), Lee et al (2018), Carta et al (2019a, Garau et al (2019), Carta et al (2020) and to design cloaking devices to hide the presence of an object in continuous and discrete elastic systems. Concerning the latter, see the studies conducted by Brun et al (2012) and Garau et al (2019).…”

“…1(b) is linked to the non-reciprocity of the system due to the gyroscopic effect, which is formally demonstrated in the present paper. The idea of connecting gyroscopic spinners to a discrete system in order to generate unidirectional edge or interfacial waves has been proposed by Nash et al (2015), Wang et al (2015), Garau et al (2018), Mitchell et al (2018), Lee et al (2018) and Garau et al (2019) for hexagonal and kagome configurations. In the present paper, the mass points are arranged in a triangular lattice pattern and the attention is focused on Rayleigh surface waves.…”