Symmetry breaking and topology of acoustic waves in phononic structures

We present examples of phononic structures that beak four types of symmetry, namely timereversal 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.