nLab anyonic braiding in momentum space -- references

Anyons in momentum-space

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

• Junyeong Ahn, Sungjoon Park, Bohm-Jung Yang, Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle, Phys. Rev. X 9 (2019) 021013 $[$doi:10.1103/PhysRevX.9.021013, arXiv:1808.05375$]$

  “here are band crossing points, henceforth called vortices”

• QuanSheng Wu, Alexey A. Soluyanov, Tomáš Bzdušek, Non-Abelian band topology in noninteracting metals, Science 365 (2019) 1273-1277 $[$arXiv:1808.07469, doi:10.1126/science.aau8740$]$

  “fundamental group of complement of nodal points/lines considered above (3)”

• Apoorv Tiwari, Tomáš Bzdušek, Non-Abelian topology of nodal-line rings in PT-symmetric systems, Phys. Rev. B 101 (2020) 195130 $[$doi:10.1103/PhysRevB.101.195130$]$

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

• Adrien Bouhon, QuanSheng Wu, Robert-Jan Slager, Hongming Weng, Oleg V. Yazyev, Tomáš Bzdušek, Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe, Nature Physics 16 (2020) 1137-1143 $[$arXiv:1907.10611, doi:10.1038/s41567-020-0967-9$]$

“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

• Haedong Park, Stephan Wong, Xiao Zhang, and Sang Soon Oh, Non-Abelian Charged Nodal Links in a Dielectric Photonic Crystal, ACS Photonics 8 (2021) 2746–2754 [doi:10.1021/acsphotonics.1c00876]

• Siyu Chen, Adrien Bouhon, Robert-Jan Slager, Bartomeu Monserrat, Non-Abelian braiding of Weyl nodes via symmetry-constrained phase transitions (formerly: Manipulation and braiding of Weyl nodes using symmetry-constrained phase transitions), Phys. Rev. B 105 (2022) L081117 $[$arXiv:2108.10330, doi:10.1103/PhysRevB.105.L081117$]$

  “Our work opens up routes to readily manipulate Weyl nodes using only slight external parameter changes, paving the way for the practical realization of reciprocal space braiding.”

• Bo Peng, Adrien Bouhon, Robert-Jan Slager, Bartomeu Monserrat, Multi-gap topology and non-Abelian braiding of phonons from first principles, Phys. Rev. B 105 (2022) 085115 [arXiv:2111.05872, doi:10.1103/PhysRevB.105.085115]

  (analog realization in phononic crystals)

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 13 423 (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]

See also:

• Robert-Jan Slager, Adrien Bouhon, Fatma Nur Ünal, Floquet multi-gap topology: Non-Abelian braiding and anomalous Dirac string phase, Nature Communications 15 1144 (2024) [arXiv:2208.12824, doi:10.1038/s41467-024-45302-2]

• 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 $ZrTe_5$ with in-plane ferromagnetism under pressure. […] one can expect that the braiding of nodes can be achieved in 2D bilayer $ZrTe_5$ 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:

• Biao Lian, Cumrun Vafa, Farzan Vafa, Shou-Cheng Zhang, Chern-Simons theory and Wilson loops in the Brillouin zone, Phys. Rev. B 95 (2017) 094512 [doi:10.1103/PhysRevB.95.094512]