Contents

# Contents

## Idea

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).

## References

### General

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

• 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 modal points in the Brillouin torus of 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 $\mathcal{C}_2\mathcal{T}$ 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:

• Jian Kang, Oskar Vafek, Non-Abelian Dirac node braiding and near-degeneracy of correlated phases at odd integer filling in magic angle twisted bilayer graphene, Phys. Rev. B 102 (2020) 035161 $[$arXiv:2002.10360, doi:10.1103/PhysRevB.102.035161$]$

• Bin Jiang, Adrien Bouhon, Zhi-Kang Lin, Xiaoxi Zhou, Bo Hou, Feng Li, Robert-Jan Slager, Jian-Hua Jiang Experimental observation of non-Abelian topological acoustic semimetals and their phase transitions, Nature Physics 17 (2021) 1239-1246 $[$arXiv:2104.13397, doi:10.1038/s41567-021-01340-x$]$

(analog realization in phononic crystals)

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.

• Bo Peng, Adrien Bouhon, Bartomeu Monserrat, Robert-Jan Slager,Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates, Nature Communications volume 13, Article number: 423 (2022) (doi:10.1038/s41467-022-28046-9)

(analog realization in phononic crystals)

it is possible to controllably braid Kagome band nodes in monolayer $\mathrm{Si}_2 \mathrm{O}_3$ using strain and/or an external electric field.

• Haedong Park, Wenlong Gao, Xiao Zhang, Sang Soon Oh, Nodal lines in momentum space: topological invariants and recent realizations in photonic and other systems, Nanophotonics 11 11 (2022) 2779–2801 $[$doi:10.1515/nanoph-2021-0692$]$

(analog realization in photonic crystals)

• Adrien Bouhon, Robert-Jan Slager, Multi-gap topological conversion of Euler class via band-node braiding: minimal models, PT-linked nodal rings, and chiral heirs $[$arXiv:2203.16741$]$