Rotational modes in a phononic crystal with fermion-like behavior

Abstract

The calculated band structure of a two-dimensional phononic crystal composed of stiff polymer inclusions in a soft elastomer matrix is shown to support rotational modes. Numerical calculations of the displacement vector field demonstrate the existence of modes whereby the inclusions and the matrix regions between inclusions exhibit out of phase rotations but also in phase rotations. The observation of the in-phase rotational mode at low frequency is made possible by the very low transverse speed of sound of the elastomer matrix. A one-dimensional block-spring model is used to provide a physical interpretation of the rotational modes and of the origin of the rotational modes in the band structure. This model is analyzed within Dirac formalism. Solutions of the Dirac-like wave equation possess a spinor part and a spatio-temporal part. The spinor part of the wave function results from a coupling between the senses (positive or negative) of propagation of the wave. The wave-number dependent spinor-part of the wave function for two superposed waves can impose constraints on the integral of the spatio-temporal part that are reflected in a fermion-like lifting of degeneracy in the phonon band structure associated with in-phase rotations.

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

“…Recently, extrinsic topological phononic crystals have demonstrated the astonishing property of non-reciprocity and backscattering-immune edge states and bulk states establishing classical equivalents of topological electronic insulators. The non-conventional topology of elastic waves in an intrinsic topological phononic structure has been associated with the notion of duality in the quantum statistics of phonons (i.e., boson vs. fermion) [4,5]. In the current paper, we illustrate the topological properties of elastic waves in the two classes of topological phononic structures.…”

“…For instance, at k = 0, the antisymmetric mode is represented by a standing wave which enforces a strict relation between the amplitude of a forward propagating wave and a backward propagating wave. This characteristic was shown [4,5] to be representative of Fermion-like behavior of phonons. As k Ñ 8 , ω Ñ`βk , the first two terms in Equation (5) go to zero and only one direction of propagation (backward) is supported by the medium (third and fourth terms in Equation (5)).…”

“…Topological electronic [1], electromagnetic [2,3], and phononic crystals [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] all have demonstrated unusual topologically constrained properties. In phononic crystals and acoustic metamaterials, symmetry breaking is linked to constraints on the topological form of acoustic wave functions.…”

“…There exists two-classes of phonon structures possessing non-conventional topology, namely intrinsic and extrinsic systems. Time-reversal symmetry in intrinsic systems [4][5][6][7][8][9][10][11][12][13][14] is broken through internal resonance or symmetry breaking structural features (e.g., chirality) and without addition of energy from the outside. Energy is added to extrinsic topological systems to break time reversal symmetry [15][16][17][18][19][20].…”

Geometric phase and topology of elastic oscillations and vibrations in model systems: Harmonic oscillator and superlattice

“…Phononic structures have also been shown recently to possess non-conventional topology as well as topologically constrained propagative properties. These properties have been achieved by breaking time-reversal symmetry through internal resonance or symmetry breaking structural features (e.g., chirality) [9][10][11][12][13][14][15][16][17][18][19] and without addition of energy from the outside. Energy can also be added to extrinsic topological elastic systems to break time reversal symmetry.…”