One-Dimensional Mass-Spring Chains Supporting Elastic Waves with Non-Conventional Topology

Abstract

There are two classes of phononic structures that can support elastic waves with non-conventional topology, namely intrinsic and extrinsic systems. The non-conventional topology of elastic wave results from breaking time reversal symmetry (T-symmetry) of wave propagation. In extrinsic systems, energy is injected into the phononic structure to break T-symmetry. In intrinsic systems symmetry is broken through the medium microstructure that may lead to internal resonances. Mass-spring composite structures are introduced as metaphors for more complex phononic crystals with non-conventional topology. The elastic wave equation of motion of an intrinsic phononic structure composed of two coupled one-dimensional (1D) harmonic chains can be factored into a Dirac-like equation, leading to antisymmetric modes that have spinor character and therefore non-conventional topology in wave number space. The topology of the elastic waves can be further modified by subjecting phononic structures to externally-induced spatio-temporal modulation of their elastic properties. Such modulations can be actuated through photo-elastic effects, magneto-elastic effects, piezo-electric effects or external mechanical effects. We also uncover an analogy between a combined intrinsic-extrinsic systems composed of a simple one-dimensional harmonic chain coupled to a rigid substrate subjected to a spatio-temporal modulation of the side spring stiffness and the Dirac equation in the presence of an electromagnetic field. The modulation is shown to be able to tune the spinor part of the elastic wave function and therefore its topology. This analogy between classical mechanics and quantum phenomena offers new modalities for developing more complex functions of phononic crystals and acoustic metamaterials.

Topological edge modes by smart patterning

“…Given an element, h, of the C * -algebra and a continuous function ϕ : C → C, one can define a functional calculus ϕ(h) by approximating ϕ by polynomials and taking the limit with respect to (20). This limit exists if and only if ϕ is continuous on the spectrum of h (see [56]).…”

Non-separable states in a bipartite elastic system

“…We have shown in reference [10][11][12], that the spinor part of the wave function when projected on the orthonormal basis 1 0 and 0 1 represents a superposition of states in the possible directions of propagation of the wave. In such a directional representation, these states are quasi-standing waves.…”

“…One can establish a one-to-one correspondence between a measurable quantity, namely the transmission coefficient along the chains and the components of the spinor part of the wave function. [10][11][12] While the states of the bipartite mechanical system are separable in a spectral representation, the possibility of measuring the spinor part of its wave function dictates another partitioning into two identical subsystems, each composed of a single elastic chain coupled to a rigid substrate. The states of elastic waves in these subsystems can also be described via a Dirac-like equation and possess 2x1 spinor amplitudes.…”

Spacetime representation of topological phononics

“…For example, elastic waves in an intrinsic topological structure composed of a one-dimensional (1D) harmonic crystal with masses attached to a rigid substrate through harmonic springs have been shown to obey a Dirac-like equation and to possess a spin-like topology [3,4]. Extrinsic topological phononic structures have been created by applying a periodic spatial modulation of the stiffness of a 1D elastic medium such that its directed temporal evolution breaks time-reversal and parity symmetries [5][6][7][8][9][10][11].…”