nLab anyonic braiding in momentum space -- references

Anyons in momentum-space

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:

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 October 25, 2023 at 15:59:34. See the history of this page for a list of all contributions to it.