Topology

Topology – When sound waves propagate in media under symmetry breaking conditions, they may exhibit amplitudes Ak=Aoeiϕ(k) that acquire a geometric phase ϕ leading to non-conventional topology. Broken symmetry phenomena lead to the concept of symmetry protected topological order. Topological acoustic waves promise designs and new device functionalities for acoustic systems that are unique, robust and avoid the loss of coherence.

Sound-supporting media with broken symmetry are a new frontier for exploring topological order. Topological electronic [1], electromagnetic [2,3] and phononic crystals [4,5] all have the astonishing property of unidirectional, backscattering-immune edge states and non-reciprocity associated with the non-conventional topology of their edge states. The observation of topologically protected propagation has to date been limited to the edges of these material. The highly desirable property of non-reciprocal and topologically protected propagation of waves inside a bulk medium has just been glimpsed in our laboratories [6]. Spatio-temporal modulation of the properties of sound-supporting media is now emerging as a universal way to achieve one-way wave propagation within the bulk. The non-conventional topology of waves in time-dependent spatially modulated media results from breaking the time-reversal symmetry. A comprehensive representation of topological waves includes the symmetry-breaking interaction into the wave function. Its amplitude is supported by a manifold in wave vector space with a non-conventional topology, which manifests itself in the form of the accumulation of a non-vanishing geometric Berry phase along closed paths (non-zero Chern number) [7,8].

Topology: Spatio-temporal laser-induced modulation (A) of the elastic properties of a chalcogenide glass (B) produces a time-dependent super-lattice due to the giant photoelastic effect. The elastic band structure (C) is asymmetric due to the breaking of time-reversal symmetry. (D) Conventional transmission (black) drops as defect mass increases due to backscattering. Forward transmission in time-dependent super-lattice is negligible (red), and backward transmission is immune to backscattering (blue) due to non-reciprocity in propagation [12].

  1. M.Z. Hasan and C.L. Kane, “Colloquium: Topological insulators,” Rev. Mod. Phys. 82, 3045 (2010).
  2. A.B. Khanikaev, S.H. Mousavi, W-K. Tse, M. Kargarian, A.H. MacDonald and G. Shvets, “Photonic topological insulators,” Nature Materials 12, 233 (2013).
  3. F.D.M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
  4. Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong and B. Zhang, “Topological Acoustics,” Phys. Rev. Lett. 114, 114301 (2015).
  5. P. Wang, L. Lu and K. Bertoldi, “Topological Phononic Crystals with One-Way Elastic Edge Waves,” submitted to Phys. Rev. E, arXiv:1504.01374 [cond-mat.mes-hall] (2015).
  6. N. Swinteck, S. Matsuo, K. Runge, J.O. Vasseur, P. Lucas and P.A. Deymier, “Bulk elastic waves with unidirectional backscattering-immune topological states in a time-dependent superlattice,” submitted to J. Appl. Phys.
  7. M.V. Berry, “Quantal Phase Factors Accompanying Adiabatic Changes,” Proc. of the Royal Soc. A 392, 45 (1984).
  8. J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62, 2747 (1989).