## Abstract

When sound waves propagate in media under symmetry breaking conditions, they may exhibit amplitudes A(k) = A_{0}e^{ iη(k)} that acquire a geometric phase η leading to non-conventional topology. In the previous chapter, we considered the consequences of breaking inversion symmetry in discrete superlattices. Here, we present examples of phononic structures that break four types of symmetry, namely time-reversal symmetry, parity symmetry, chiral symmetry and particle-hole symmetry. The implications of symmetry breaking on the topology of the acoustic wave function in the space of its Eigen values are discussed. Particular attention is focused on the torsional topology of acoustic waves in periodic media in wave vector space. Two types of approach to achieve symmetry breaking are considered: (a) intrinsic topological phononic structures whereby symmetry breaking occurs from the internal structural characteristics, and (b) extrinsic topological phononic structures where external stimuli such as spatio-temporal modulations of the physical properties of the medium are used to break symmetry. Broken symmetry phenomena lead to the concept of symmetry protected topological order. Broken symmetry also underpins the concepts of non-reciprocal wave propagation. Topological acoustic waves promise designs and new device functionalities for acoustic systems that are unique, robust and avoid the loss of coherence. Furthermore, the self-interaction of an elastic wave in topological phononic structures creates states determined by self-interference phenomena, revealing a quantum mechanical analogy with the concept of particle–wave duality. In the phonon representation of sound and elastic waves, these self-interference phenomena uncover the notion of duality in the quantum statistics (i.e., boson vs. fermion characterized by the symmetry of multiple particle states). Furthermore, we also consider the partitioning the phononic structures presented in this chapter into subsystems and the separability and non-separability of their wave functions into tensor products. Separability is shown to be relative to the choice of the subsystems in which one partitions the system of interest. The choice of the subsystems in turn may be dictated by the possible observables and measurements. The non-separability of these phononic structures is analogous to the quantum phenomenon of entanglement. Analogies with quantum phenomena interrogate mechanical waves in ways thought to be reserved for the microscopic realm.

Original language | English (US) |
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Title of host publication | Springer Series in Solid-State Sciences |

Publisher | Springer Science and Business Media Deutschland GmbH |

Pages | 81-161 |

Number of pages | 81 |

DOIs | |

State | Published - 2017 |

### Publication series

Name | Springer Series in Solid-State Sciences |
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Volume | 188 |

ISSN (Print) | 0171-1873 |

ISSN (Electronic) | 2197-4179 |

## ASJC Scopus subject areas

- Condensed Matter Physics