Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe


Weyl semimetals in three-dimensional crystals provide the paradigm example of topologically protected band nodes. It is usually taken for granted that a pair of colliding Weyl points annihilate whenever they carry opposite chiral charge. In stark contrast, here we report that Weyl points in systems that are symmetric under the composition of time reversal with a π rotation are characterized by a non-Abelian topological invariant. The topological charges of the Weyl points are transformed via braid phase factors, which arise upon exchange inside symmetric planes of the reciprocal momentum space. We elucidate this process with an elementary two-dimensional tight-binding model that is implementable in cold-atom set-ups and in photonic systems. In three dimensions, interplay of the non-Abelian topology with point-group symmetry is shown to enable topological phase transitions in which pairs of Weyl points may scatter or convert into nodal-line rings. By combining our theoretical arguments with first-principles calculations, we predict that Weyl points occurring near the Fermi level of zirconium telluride carry non-trivial values of the non-Abelian charge, and that uniaxial compression strain drives a non-trivial conversion of the Weyl points into nodal lines.

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Fig. 1: Reciprocal braiding of band nodes.
Fig. 2: Elementary protocol for braiding band nodes.
Fig. 3: Frame-rotation charge.
Fig. 4: Berry curvature versus Euler form.
Fig. 5: Conversion of Weyl points in 3D momentum space.
Fig. 6: Weyl points and nodal lines in ZrTe.

Data availability

All other data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request. Input files used to generate the first-principles data are provided in the Materials Cloud Archive71.

Code availability

The Mathematica notebook for computing the Euler class of a collection of band nodes by implementing the method presented in Supplementary Section H is available online65.


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We acknowledge valuable discussions with C.C. Wojcik, A. Vishwanath and B.A. Bernevig. R.-J.S. acknowledges funding via A. Vishwanath from the Center for Advancement of Topological Semimetals, an Energy Frontier Research Center funded by the US Department of Energy Office of Science, Office of Basic Energy Sciences, through the Ames Laboratory under contract no. DE-AC02-07CH11358, the Marie Skłodowska-Curie programme under EC grant no. 842901, and Trinity College as well as the Winton programme at the University of Cambridge. T.B. was supported by the Gordon and Berry Moore Foundation’s EPiQS Initiative, grant no. GBMF4302, and by the Ambizione Program of the Swiss National Science Foundation, grant no. 185806. Q.-S.W. and O.V.Y. acknowledge support from NCCR Marvel. H.M.W. acknowledges support from the Ministry of Science and Technology of China under grants 2018YFA0305700 and 2016YFA0300600, the Strategic Priority Research Program of Chinese Academy of Sciences (grant no. XDB33000000) and the K. C. Wong Education Foundation (GJTD-2018-01). First-principles calculations have been performed at the Swiss National Supercomputing Centre (CSCS) under project no. s832 and s1008 and the facilities of Scientific IT and Application Support Center of EPFL.

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A.B., R.-J.S. and T.B. contributed equally to the theoretical analysis in this work and wrote the manuscript. Q.-S.W. discovered the nodal conversion in ZrTe and obtained the presented first-principles data. H.M.W. and O.V.Y. were involved in the discussion and analysis of the first-principles data. All authors discussed and commented on the manuscript.

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Correspondence to Adrien Bouhon or QuanSheng Wu or Robert-Jan Slager.

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Extended data

Extended Data Fig. 1 Wilson loop computation of Euler class of two principal nodes in the presence of two adjacent nodes.

a-b Two possible flows of a base loop (lν, ν\(\in\) [0, 1]) over the punctured Brillouin zone containing two principal nodes (black) and excluding two adjacent nodes (grey). The flow starts at a base point (l0 = x0) and ends at the boundary \({l}_{1}=\partial {\rm{BZ}}-\partial {{\mathcal{D}}}^{\epsilon }\), with the orange arrow indicating the direction of the flow. Assuming that the pair of principal nodes were created first (along the dashed line between them), the Euler class is \(\chi [{\rm{BZ}}-{{\mathcal{D}}}^{\epsilon }]=0\) in case a, and \(\chi [{\rm{BZ}}-{{\mathcal{D}}}^{\epsilon }]=1\) in case b. c-f Wilson loop (c,d) and Pfaffian (e,f) as a function of the flow of base loop over the punctured Brillouin zone, that is corresponding to a for (c,e), and to b for (d,f). e The zero winding of the Pfaffian, Δζ = 0, indicates a trivial frame-rotation charge of the two principal nodes (that is zero Euler class on \({\rm{BZ}}-{{\mathcal{D}}}^{\epsilon }\)). f The non-zero winding of the Pfaffian, Δζ/(2π) = 1, indicates a non-trivial frame-rotation charge of the two principal nodes (that is non-vanishing Euler class on \({\rm{BZ}}-{{\mathcal{D}}}^{\epsilon }\)).

Extended Data Fig. 2 Fermi surface and band structures of ZrTe with and without strain.

a. Fermi surface of ZrTe under ambient conditions, with pairs of Weyl nodes of opposite chirality located inside the cyan pockets close to kz = 0 plane. b and c. Band structures of ZrTe along high-symmetry lines of the Brillouin under 0% resp. under 2.6% uniaxial compression strain. Path M-K-Γ lies within the kz = 0 mirror-invariant plane. The mirror eigenvalues + i and − i are indicated by the red vs. blue colour of the corresponding energy band. The plotted data were obtained with PBE+HSE06 functional.

Extended Data Fig. 3 Band structure of ZrTe.

Band structure of ZrTe as computed from PBE+HSE06 (left) resp. from PBE (right) functional.

Extended Data Fig. 4 Comparison of principal band nodes of ZrTe and MoP.

Band structure and location of nodal points of ZrTe and MoP, obtained with PBE+HSE06 functional. The band nodes in panels c and d correspond to degeneracies of energy bands marked with green and red colour in panels a and b.

Extended Data Fig. 5 Band structure and principal band nodes of ambient TaAs and with strain.

Band structure and band node locations in the first Brillouin zone of TaAs ad under ambient conditions, and eh with 5% [001] uniaxial strain obtained with PBE functional. The first column shows the band structure along the ΓX line. The next three columns show the location of band nodes in 3-dimension view, top view, and front view, respectively. Weyl points with chirality +1 and -1 are respectively indicated with magenta and green spheres. Nodal lines of the strained TaAs are indicated as brown lines. The \({C}_{2}{\mathcal{T}}\)-invariant plane is indicated with cyan colour, while the vertical mirror planes are displayed in shades of yellow.

Extended Data Fig. 6 Euler curvature in TaAs.

Euler curvature of a pair of Weyl points inside the \({C}_{2}{\mathcal{T}}\)-invariant plane of TaAs under ambient conditions. For a discussion, see Methods.

Extended Data Fig. 7 Band structure of nine WC-type materials obtained with PBE functional.

The corresponding nodal structure near the K-point under ambient conditions is indicated in each panel.

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Bouhon, A., Wu, Q., Slager, R. et al. Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe. Nat. Phys. (2020).

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