added pointer to:

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

with this quote:

]]>“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.”

added pointer to this recent article:

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

added a warning:

Beware that various authors consider braids/knots formed by nodal

linesin3dsemimetals, i.e. knotted nodal lines in 3 spatial dimensions, as opposed to worldlines of nodal points in 2+1 spacetime dimensions needed for anyon-braiding as considered above.

and then, to start with, pointer to this 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]

added pointer to today’s

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

also pointer to:

- 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]

which stands out as discussion the phenomen for (twisted bilayer-) graphene

]]>added pointer to:

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

with this quote:

]]>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

I’d also like to compile a list of references which admit or at least indicate that the usual hope of having anyonic topological order on a toroidal (or even higher genus) *position space* is dubious.

I had come across several such references in the past, but didn’t record them in this respect. So now I need to re-compile a list. For the moment I have

which is quite explicit.

I vividly remember a reference on the alleged classification of topological order by FQFT/cobordism which tried to argue that crystal impurities could mimic higher genus position space geometries. But it seems I have lost it and now it’s the proverbial needle in the haystack.

]]>added more items and more quotes

]]>a bare list of references, to be `!include`

-ed into the References-section of relevant entries (such as at *braid group representation* and at *semi-metal*).

Had originally compiled this list already last April (for this MO reply) but back then the nLab couldnt be edited

]]>