nLab phononic crystal




physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics

Solid state physics



A phononic crystal or acoustic crystal is a material with periodic elastic properties in which sound? waves (ultimately: phonons) behave much like Bloch waves of electrons inside an actual crystal, such as in that they form energy bands separated by band gaps (KHDD93).

By suitably engineering phononic crystals they may emulate various phenomena seen elsewhere in nature or not seen anywhere else, whence one also speaks of “meta-materials”.

In much the same way, there are phononic crystals.

For instance, various aspects of topological phases of matter (such as Berry phases, see Liu, Chen & Xu 2020; or phononic semimetals, see Li et al. 2022) have analogs realizations in phononic crystals (“topological phononics”, see the references below). Notably braiding of band nodes has been demonstrated in phononic crystals (Peng, Bouhon, Monserrat & Slager 2022, Peng, Bouhon, Slager & Monserrat 2022) and similarly in photonic semimetals (Park, Gao, Zhang & Oh 2022).



Early articles on acoustic energy band structure in phononic crystals:

  • M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani, Acoustic band structure of periodic elastic composites, Phys. Rev. Lett. 71 2022 (1993) [doi:10.1103/PhysRevLett.71.2022]

  • M. M. Sigalas, E. N. Economou, Elastic and acoustic wave band structure, Journal of Sound and Vibration 158 2 (1992) 377-382 [doi:10.1016/0022-460X(92)90059-7]

Review of phononic crystals:

  • Ming-Hui Lu, Liang Feng, Yan-Feng Chen, Phononic crystals and acoustic metamaterials, Materials Today 12 12 (2009) 34-42 [doi:10.1016/S1369-7021(09)70315-3]

  • Yizhou Liu, Xiaobin Chen, Yong Xu, Topological Phononics: From Fundamental Models to Real Materials, Advanced Functional Materials 30 8 (2020) [doi:10.1002/adfm.201904784]

Further resources:

Topological phononics

On topological phononics:

  • S. Huber, Topological mechanics, Nature Phys 12* 621–623 (2016) [doi:10.1038/nphys3801]

  • J. Li, J. Liu, S.A. Baronett, Computation and data driven discovery of topological phononic materials, Nat Commun 12 1204 (2021) [doi:10.1038/s41467-021-21293-2]

  • Yan Du, Weiguo Wu, Wei Chen, Yongshui Lin, Qingjia Chi: Control the structure to optimize the performance of sound absorption of acoustic metamaterial: A review, AIP Advances 11 060701 (2021) [doi:10.1063/5.0042834]

The phononic ananlog of topological insulators/quantum Hall effect:

  • Roman Süsstrunk, Sebastian D. Huber, Observation of phononic helical edge states in a mechanical ‘topological insulator’, Science 349 47 (2015) [arXiv:1503.06808]

Anyons in momentum-space

On non-trivial braiding of nodal points in the Brillouin torus of topological semi-metals (“braiding in momentum space”):

“a new type non-Abelian ‘braiding’ of nodal-line rings inside the momentum space”

“Here we report that Weyl points in three-dimensional (3D) systems with 𝒞 2𝒯\mathcal{C}_2\mathcal{T} symmetry carry non-Abelian topological charges. These charges are transformed via non-trivial phase factors that arise upon braiding the nodes inside the reciprocal momentum space.”

Braiding of Dirac points in twisted bilayer graphene:

Here, we consider an exotic type of topological phases beyond the above paradigms that, instead, depend on topological charge conversion processes when band nodes are braided with respect to each other in momentum space or recombined over the Brillouin zone. The braiding of band nodes is in some sense the reciprocal space analog of the non-Abelian braiding of particles in real space.


we experimentally observe non-Abelian topological semimetals and their evolutions using acoustic Bloch bands in kagome acoustic metamaterials. By tuning the geometry of the metamaterials, we experimentally confirm the creation, annihilation, moving, merging and splitting of the topological band nodes in multiple bandgaps and the associated non-Abelian topological phase transitions

new opportunities for exploring non-Abelian braiding of band crossing points (nodes) in reciprocal space, providing an alternative to the real space braiding exploited by other strategies.

Real space braiding is practically constrained to boundary states, which has made experimental observation and manipulation difficult; instead, reciprocal space braiding occurs in the bulk states of the band structures and we demonstrate in this work that this provides a straightforward platform for non-Abelian braiding.

See also:

  • Robert-Jan Slager, Adrien Bouhon, Fatma Nur Ünal, Floquet multi-gap topology: Non-Abelian braiding and anomalous Dirac string phase [arXiv:2208.12824]

  • Huahui Qiu et al., Minimal non-abelian nodal braiding in ideal metamaterials, Nature Communications 14 1261 (2023) [doi:10.1038/s41467-023-36952-9]

  • Wojciech J. Jankowski, Mohammedreza Noormandipour, Adrien Bouhon, Robert-Jan Slager, Disorder-induced topological quantum phase transitions in Euler semimetals [arXiv:2306.13084]

  • Seung Hun Lee, Yuting Qian, Bohm-Jung Yang, Euler band topology in spin-orbit coupled magnetic systems [arXiv:2404.16383]

    “Based on first-principles calculations, we report that such nodal point braiding in 2D electronic bands can be realized in a MSWI candidate, the bilayer ZrTe 5ZrTe_5 with in-plane ferromagnetism under pressure. […] one can expect that the braiding of nodes can be achieved in 2D bilayer ZrTe 5ZrTe_5 under the influence of an external in-plane Zeeman field.”

Incidentally, references indicating that the required toroidal (or yet higher genus) geometry for anyonic topological order in position space is dubious (as opposed to the evident toroidal geometry of the momentum-space Brillouin torus): Lan 19, p. 1, ….

Knotted nodal lines in 3d semimetals

Beware that various authors consider braids/knots formed by nodal lines in 3d semimetals, i.e. knotted nodal lines in 3 spatial dimensions, as opposed to worldlines (in 2+1 spacetime dimensions) of nodal points in effectively 2d semimetals needed for the anyon-braiding considered above.

An argument that these nodal lines in 3d space, nevertheless, may be controlled by Chern-Simons theory:

Last revised on September 12, 2022 at 10:42:58. See the history of this page for a list of all contributions to it.