Non-Abelian reciprocal braiding of Weyl points and its manifestation in $\textrm{ZrTe}$

TL;DR

This paper uncovers a non-Abelian topological invariant in certain Weyl semimetals, revealing novel behaviors of Weyl points under symmetry constraints and predicting their transformation in ZrTe under strain.

Contribution

It introduces the concept of non-Abelian topological charges for Weyl points in symmetric systems and demonstrates their effects through models and first-principles calculations.

Findings

Weyl points in specific symmetric systems are characterized by non-Abelian invariants.

Exchange of Weyl points leads to braid phase factors affecting their topological charge.

Uniaxial strain in ZrTe causes Weyl points to convert into nodal lines.

Abstract

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 a stark contrast, here we report that Weyl points in systems symmetric under the composition of time-reversal with a $\pi$-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 implementable in cold-atoms setups 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 ($\textrm{ZrTe}$) 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.

Figures

Equations

$$\text{Eu}(\mathbf{k})=⟨\nabla u^1(\mathbf{k})|\times |\nabla u^2(\mathbf{k})⟩,$$

$$\mathrm{d}\Omega =\mathbf{n}\cdot \left({\partial }_{k_1}\mathbf{n}\times {\partial }_{k_2}\mathbf{n}\right)\mathrm{d}{k}_1\mathrm{d}{k}_2$$

$$\chi (\mathcal{D})=\frac{1}{2\pi }\left[{\int }_{\mathcal{D}}\text{Eu}(\mathbf{k})\mathrm{d}{k}_1\mathrm{d}{k}_2-{\oint }_{\partial \mathcal{D}}\text{a}(\mathbf{k})\cdot \mathrm{d}\mathbf{k}\right],$$

$$H(\boldsymbol{k};t)=\left(\begin{array}[]{ccc}f(t)&g(\boldsymbol{k})&g^{*}(\boldsymbol{k})\\ g^{*}(\boldsymbol{k})&0&h(\boldsymbol{k};t)\\ g(\boldsymbol{k})&h^{*}(\boldsymbol{k};t)&0\end{array}\right)$$

$$\mathcal{A}(\mathbf{k})={\mathfrak{U}}^{†}(\mathbf{k})d\mathfrak{U}(\mathbf{k}),$$

$${\mathcal{A}}_i^{ab}(\mathbf{k})=⟨u^a(\mathbf{k})|{\partial }_{k_i}{u}^b(\mathbf{k})⟩,$$

$$\tilde{\mathcal{A}}=X^{†}\mathcal{A}X+X^{†}dX,$$

$$\mathcal{F}=d\mathcal{A}+\mathcal{A}\wedge \mathcal{A},$$

$${\mathcal{F}}_{ij}^{ab}$$

$$\tilde{\mathcal{F}}=X^{†}\mathcal{F}X,$$

$${\mathsf{F}}_{ij}=-\mathrm{i}{\mathcal{F}}_{ij}^{aa}=-\mathrm{i}\left(⟨{\partial }_{k_i}{u}^a|{\partial }_{k_j}{u}^a⟩-⟨{\partial }_{k_j}{u}^a|{\partial }_{k_i}{u}^a⟩\right)$$

$$\text{a}:=\mathrm{Pf}({\mathcal{A}}_i)d{k}^i$$

$$\tilde{\text{a}}$$

$$\mathrm{Pf}(X^{\top }\text{mathpzc}MX)=\det (X)\mathrm{Pf}(\text{mathpzc}M).$$

$$\text{Eu}:=d\mathrm{a}\text{(for n=2 bands),}$$

$$\text{Eu}=\mathrm{Pf}(d\mathcal{A})=\mathrm{Pf}(\mathcal{F}-\mathcal{A}\wedge \mathcal{A})=\mathrm{Pf}(\mathcal{F})\text{(for n=2 bands),}$$

$$\text{Eu}:=\mathrm{Pf}({\mathcal{F}}_{ij})d{k}^i\wedge d{k}^j.\text{(for arbitrary n),}$$

$$\widetilde{\mathrm{Eu}}=\det (X)\mathrm{Eu},$$

$$\text{Eu}=\left(⟨{\partial }_{k_x}{u}^1|{\partial }_{k_y}{u}^2⟩-⟨{\partial }_{k_y}{u}^1|{\partial }_{k_x}{u}^2⟩\right)d{k}^x\wedge d{k}^y,$$

$$\chi (E)=\frac{1}{2\pi }{\oint }_B\text{Eu}\text{(for n=2 bands)}$$

$$c_1(E)=\frac{1}{2\pi }{\oint }_B\mathsf{F}$$

$$2\pi \chi (E)$$

$$m\in \text{mathbbold}Z$$

$${\mathsf{F}}_{ij}$$

$$=$$

$$\mathsf{F}_{ij}=\left<{\partial_{k_{i}}u^{1}}|{\partial_{k_{j}}u^{2}}\right>-\left<{\partial_{k_{i}}u^{2}}|{\partial_{k_{j}}u^{1}}\right>\Big{)}+\textrm{c.c.}=\textrm{Eu}_{ij}$$

$$\textrm{Eu}\bigg{[}\left|{u^{1}}\right>,\left|{u^{2}}\right>\bigg{]}=\mathsf{F}\bigg{[}\tfrac{1}{\sqrt{2}}\left(\left|{u^{1}}\right>+\mathrm{i}\left|{u^{2}}\right>\right)\bigg{]}$$

$${\mathcal{W}}_l=\exp \left\{{\oint }_l\mathcal{A}(\mathbf{k})\cdot d\mathbf{l}(\mathbf{k})\right\},$$

$$\zeta (l)=\mathrm{Pf}\left[\mathrm{i}{\mathcal{H}}_{{\mathcal{W}}_l}\right]=\mathrm{Pf}\left[\log {\mathcal{W}}_l\right].$$

$$|{\varphi }_{\alpha },\mathbf{k}⟩=\frac{1}{\sqrt{N_{\alpha }}}\sum _{{\mathbf{R}}_n}{e}^{\mathrm{i}\mathbf{k}\cdot {\mathbf{R}}_n}|w,{\mathbf{R}}_n+{\mathbf{r}}_{\alpha }⟩,\alpha =A,B,C,$$

Full text

The robust and illustrious properties of topological order, such as protected edge states and the possibility of excitations that exhibit non-trivial braiding statistics Wen95_top, open up routes to potentially translate mathematical understanding of the physical phenomena to a new generation of quantum technology. This has arguably also fuelled the discovery of topological band structures HasanKane10_RMP; Qi11_RMP that can effectively give rise to such physical features HasanKane10_RMP; Qi11_RMP; Majorana. The past decade has witnessed considerable progress in cataloguing topological insulators and semimetals Kitaev09_AIP; Ryu10_NJP; Fu11_PRL; Slager12_NatPhys; Bzduvsek16nodal; Fang:2016; Bzdusek:2017; Kruthoff17_PRX; Bouhon:global_top; Po17_NatCommun; Bradlyn17_Nat; Slager2019; Holler18_PRB; Zhang2019; topomat, thereby providing an increasingly viable platform for bringing the potential of topological materials to experiment. Notably, Weyl semimetals were convincingly shown to exhibit topologically protected surface Fermi arcs and chiral-anomaly-induced negative magnetoresistance Wan:2011; Weng:2015; Lv:2015; Xu:2015; Huang:2015.

Here we report that Weyl points in three-dimensional (3D) systems with $C_{2}\mathcal{T}$ symmetry (time reversal composed with a $\pi$-rotation) 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. This discovery extends the previous theoretical works on non-Abelian disclination defects in nematic liquids Poenaru:1977; Volovik:1977; Madsen:2004; Alexander:2012; Prx2016, Dirac lines in space-time-inversion symmetric metals Wu:2018b; Tiwari:2019, and Dirac points of twisted bilayer graphene Ahn:2019; Ahn:2018; Cao_2018; Ashvin_2019; Song2019; Bouhon2018_fragile. We show that interplay of the non-Abelian topology with point-group symmetry greatly enriches the range of topological phase transitions for Weyl points in 3D crystals.

Below, we illustrate the reciprocal braiding with an elementary 2D model, which is directly implementable in cold-atoms and photonic systems. We relate the descriptions of the non-Abelian topology via quaternion numbers Wu:2018b resp. Euler class Ahn:2019, and we numerically implement the latter. It is shown that the interplay of the non-Abelian charges with the chiralities of Weyl points and with mirror symmetry results in non-trivial conversions between Weyl points and nodal lines. Finally, we combine our formalism with first-principles modeling of the existing material zirconium telluride (ZrTe) and related compounds, which provide simple examples for the predicted nodal conversions.

Elementary braiding protocol

The ability of band nodes in $C_{2}\mathcal{T}$-symmetric systems to pairwise annihilate crucially depends on the presence of band nodes in other band gaps Wu:2018b; Tiwari:2019; Ahn:2019; Ahn:2018. This enables non-trivial “reciprocal braiding” inside the momentum ($\boldsymbol{k}$) space, illustrated in Fig. 1. For three-band models we introduce the following terminology. The main gap of interest is called “principal gap”. Accordingly, band nodes in this gap are described as “principal nodes”. The other band gap, as well as the corresponding band and nodes, are called “adjacent”.

As an illustrative model for node braiding in two dimensions, we consider a three-band Hamiltonian $H(\boldsymbol{k};t)$ [see Eq. (4) of Methods], where $t\in[-10,10]$ is a tuning parameter (“time”). The practical implementation of the model requires tuning only three tight-binding parameters, namely the potential on one site, and the hopping amplitude between the other two sites along the horizontal resp. the vertical direction.

The model exhibits nodal points along the $(1\bar{1})$ and $(11)$ diagonals of the Brillouin zone (BZ). In Fig. 2c, we show snapshots of the band structure along the two diagonals during the braiding protocol with solid resp. dashed curves. At the initial time $t\!=\!-10$, the bands are energetically separated. At $t\!=\!-8$, the adjacent gap exhibits a pair of nodal points moving from $\Gamma$ (where they were created) towards M along $(1\bar{1})$. At $t\!=\!-4$, there are additional two principal nodes moving from M (where they were created) towards $\Gamma$ along $(11)$. At $t\!=\!-2$, the principal nodes meet at $\Gamma$. Remarkably, instead of annihilating, we find that the principal nodes “bounce” in the $(1\bar{1})$ direction, where they follow their adjacent counterparts, as visible at $t\!=\!0$. Fig. 2b shows the full 2D band structure at this very time. At $t\!=\!2$, the two adjacent nodes meet at M and also fail to annihilate, as can be seen at time $t\!=\!4$ where they progress towards $\Gamma$. At $t\!=\!8$, the adjacent nodes have been annihilated at $\Gamma$. Finally, at $t\!=\!10$, the principal nodes have been annihilated at M, and the bands have become fully separated again. Panel Fig. 2d displays the configuration of the nodal points at a few times, keeping track of their past trajectory.

Non-Abelian topology

The path-dependent capability of band nodes to annihilate is a consequence of underlying non-Abelian band topology Wu:2018b; Tiwari:2019; Ahn:2019; Ahn:2018. Before delving into its formal mathematical description, we attempt to visualize the non-Abelian obstruction in simple terms. Note that for two-dimensional systems $C_{2}\mathcal{T}$ symmetry implies the existence of a basis in which the Bloch Hamiltonian $H(\boldsymbol{k})$ is a real symmetric matrix [for proof see Sec. C of Supplementary Information (SI)].

For momenta where the energy bands are non-degenerate, we form an energetically ordered set of $N$ real Bloch states, $\{\left|{u^{j}(\boldsymbol{k})}\right>\}_{j=1}^{N}=:{F}(\boldsymbol{k})$, which constitutes an orthonormal frame Wu:2018b. The frame is well defined only up to the $\pm$ sign of each eigenstate, implying a gauge degree of freedom. This coincides with the gauge description of the order-parameter space of biaxial nematics Prx2016, which are known to exhibit disclination defects described by a non-Abelian group Poenaru:1977; Volovik:1977; Madsen:2004; Alexander:2012.

The constructed frames allow us to assign a frame-rotation charge to each closed path that avoids band nodes. If one varies the momentum along a closed path based at $\boldsymbol{k}_{0}$, the Hamiltonian returns to its original form. Nonetheless, the initial and the final frame at $\boldsymbol{k}_{0}$ may differ by a gauge transformation, altering the vielbein spanning the frame. Notably, such a transformation occurs if one encircles a band node. As one moves along a tight loop around the node formed by a pair of bands, then the two Bloch states describing those bands perform a $\pi$-rotation, while the other states are essentially constant.

Given two nodes inside the same band gap, one may wonder how their associated frame rotations compose. One possibility is that the second rotation undoes the first, e.g. if we rotate by $\pi$ in the reverse direction. In that case the two nodes annihilate when brought together. Alternatively, the rotations could revolve in the same direction. Although the total $2\pi$-rotation looks like a do-nothing transformation, the Dirac’s belt trick Francis:1994 reveals that a $2\pi$-rotation cannot be trivially undone (while a $4\pi$ rotation can). Mathematically, this corresponds to the non-trivial fundamental group $\pi_{1}[\mathsf{SO}(N)]=\mathbbold{Z}_{2}$ for $N>2$. Physically, this implies that a pair of nodes associated with a $2\pi$ frame rotation cannot annihilate Sjoeqvist_2004.

For the elementary braiding Hamiltonian [Eq. (4) in Methods], we study in Fig. 3 the accumulated frame-rotation angle on two paths that enclose a pair of principal nodes. We decompose the 3D rotation matrix along the path using the rotation generators $L_{x,y,z}$ as $F(\boldsymbol{k})^{\top}\cdot F(\boldsymbol{k}_{0})=\textrm{exp}[\alpha L_{x}+\beta L_{y}+\gamma L_{z}]$, and define the rotation angle $\varphi\!=\!\sqrt{\alpha^{2}+\beta^{2}+\gamma^{2}}$. As expected, we find that the total rotation angle equals [math] ($2\pi$) if the nodes can (cannot) annihilate. The difference for the two paths originates from non-commutativity of rotations, namely the rotation angle $\alpha$ acquired as one circumnavigates a principal node is reversed after conjugation with the overall $\pm\pi$ rotation associated with the adjacent node ($\mathrm{e}^{\pi L_{j}}\mathrm{e}^{\alpha L_{i}}\mathrm{e}^{\pi L_{j}}\!=\!\mathrm{e}^{-\alpha L_{i}}$ for $i\neq j$). As a result, the topological charge of principal nodes anticommutes with the topological charge of adjacent nodes. This property has been modelled by the non-Abelian quaternion group $\mathsf{Q}=\{\pm 1,\pm\mathrm{i},\pm\mathrm{j},\pm\mathrm{k}\}$ Wu:2018b (indicated in Fig. 1b) resp. using Dirac strings Ahn:2019. We show the equivalence of the two approaches in Sec. G of SI.

Euler class

While the frame rotations computed in Fig. 3 faithfully predict the ability of band nodes to annihilate, the method is computationally too costly for many-band models. Instead, to study real materials, we utilize a tool introduced by Ref. Ahn:2019 to describe a fragile-topological phase of twisted bilayer graphene near the magic angle Cao_2018; Bouhon2018_fragile; Ashvin_2019; Song2019. If $\left|{u^{1}(\boldsymbol{k})}\right>$ and $\left|{u^{2}(\boldsymbol{k})}\right>$ are real Bloch states of a pair of bands, their Euler form is Zhao:2017

[TABLE]

and the integral of Euler form over a closed surface defines integer topological invariant called Euler class Ahn:2018. Below, we recast this notion to describe stable topology of band nodes in many-band models, while also assuming an unconventional partitioning of energy bands. We first examine the geometric meaning of this mathematical object, not clarified by the earlier works Ahn:2019; Ahn:2018, and apply it to band node analysis in two dimensions. In the next section we generalize to 3D systems, and discuss the interplay of Euler class with point-group symmetry.

The concepts of Euler form and Euler class can be understood as refinements of Berry curvature and of Chern numbers (a detailed exposition appears in Methods). To reveal their relation, consider a two-band complex Hamiltonian $H(\boldsymbol{k})\!=\!\boldsymbol{h}(\boldsymbol{k})\!\cdot\!\boldsymbol{\sigma}$, where $\{\sigma_{i}\}_{i=1}^{3}$ are the Pauli matrices and $\boldsymbol{h}(\boldsymbol{k})$ is a three-component real vector. The integral of Berry curvature over an infinitesimal domain $\mathrm{d}k_{1}\mathrm{d}k_{2}$ can be expressed HasanKane10_RMP as one half of the solid angle

[TABLE]

covered by unit vector $\boldsymbol{n}(\boldsymbol{k})\!=\!\boldsymbol{h}(\boldsymbol{k})/\lVert{\boldsymbol{h}(\boldsymbol{k})}\rVert$ as $\boldsymbol{k}$ ranges over the domain, cf. Fig. 4a. If momentum ranges over a closed manifold, then $\boldsymbol{n}$ wraps around the unit sphere an integer number of times. Berry curvature thus integrates to integer multiples of $2\pi$, defining the Chern number. The theory of characteristic classes Milnor:1975 predicts that the quantization persists in models with more than two band.

In analogy, the simplest scenario with a non-trivial Euler class of a pair of bands occurs in three-band models. We find (see Fig. 4b and Sec. D of SI), that the integral of the Euler form over an infinitesimal domain $\mathrm{d}k_{x}\mathrm{d}k_{y}$ is equal to the solid angle $\mathrm{d}\Omega$ in Eq. (2) (without the $1/2$ prefactor), where $\boldsymbol{n}\!=\!\boldsymbol{u}_{1}\!\times\!\boldsymbol{u}_{2}$ is the cross product of the two Bloch states. Using arguments as before, one concludes that for three-band models the Euler form on closed manifolds integrates to integer multiples of $4\pi$. Although this simple geometric interpretation of the Euler form becomes insufficient in many-band models, it again follows from the theory of characteristic classes Milnor:1975 that the quantization to integer multiples of $2\pi$ persists, defining an integer topological invariant akin to the Chern number. We show in Methods that the Euler form of two real states $\left|{u^{1}}\right>$ and $\left|{u^{2}}\right>$ equals to the Berry curvature of complex state $\left(\left|{u^{1}}\right>{+}\mathrm{i}\left|{u^{2}}\right>\right)/\sqrt{2}$, thus enabling an efficient computational implementation Fukui:2005 (see Sec. H of SI).

Node diagnosis and band partitioning

The gauge degree of freedom may alter the overall sign of the Euler form. Consequently (see Secs. C and D of SI), $\textrm{Eu}(\boldsymbol{k})$ can be defined around principal nodes, but not around adjacent nodes. Therefore, to predict whether a pair of principal nodes annihilate when brought together along a specified trajectory, we propose the following strategy, which generalizes the methods of Ref Ahn:2019. We choose a region (disc) $\mathcal{D}$ that (i) contains the trajectory, and that (ii) does not contain any additional principal nor adjacent nodes. On the boundary $\partial\mathcal{D}$ we construct the Euler connection ${\textbf{a}}(\boldsymbol{k})\!=\!\left<{u^{1}(\boldsymbol{k})}|{\boldsymbol{\nabla}u^{2}(\boldsymbol{k})}\right>$. We define Euler class over $\mathcal{D}$,

[TABLE]

which is an integer topological invariant whenever the disc $\mathcal{D}$ contains an even number of principal nodes and no adjacent nodes (for further details see Sec. F of SI). Crucially (as elaborated in Sec. E of SI), the real Bloch states pertaining to the principal bands exhibit a singularity at principal nodes, which invalidates the Stokes’ theorem and prevents the cancellation of the two integrals in Eq. (3).

If the principal nodes inside $\mathcal{D}$ are able to collectively annihilate, then $\chi(\mathcal{D})$ must be zero. This is because annihilating all the nodes makes the Euler form exact in terms of the Euler connection, i.e. $\textrm{Eu}=\boldsymbol{\nabla}\times\textbf{a}$ in $\mathcal{D}$, in which case the Stokes’ theorem guarantees cancellation of the two integrals. Conversely, non-vanishing $\chi(\mathcal{D})$ indicates an obstruction for annihilating the principal nodes. We confirm this for the model in Eq. (4) in Sec. H of SI using a computational algorithm detailed therein.

We have also developed a complementary algorithm that allows us to find the Euler class of a collection of principal nodes using the increasingly appreciated paradigm of Wilson-loop flows, which we discuss in Methods and illustrate in Extended Data Fig. 1.

For many-band models, such as when modelling real materials, we require the two principal bands to be separated inside $\mathcal{D}$ by an energy gap on both sides, i.e. from above and from below. Such partitioning of energy bands into three groups is in contrast with the contemporary paradigm in topological band theory Kitaev09_AIP; Ryu10_NJP, which is to partition the bands as occupied vs. unoccupied via a single energy gap. This distinction explains why the stable integer invariant in Eq. (3) has been overlooked by previous works classifying the band node topology Bzdusek:2017; Fang:2016.

Non-Abelian conversions in 3D

We now shift our attention to three spatial dimensions. It is understood that $C_{2}\mathcal{T}$ symmetry can stabilize Weyl points (WPs) inside high-symmetry “$C_{2}\mathcal{T}$-planes”, as observed e.g. in the $k_{z}\!=\!0$ plane of $\textrm{WTe}_{2}$ Soluyanov:2015, MoP Lv2017 and TaAs Weng:2015; Lv:2015; Xu:2015. While such WPs are characterized by their chiral charge Wan:2011, the $C_{2}\mathcal{T}$ symmetry assigns them an extra frame-rotation charge defined by the Hamiltonian inside the symmetric plane. Importantly, these charges carry complementary pieces of topological information. While the chiral charge discloses whether a pair of WPs can annihilate, the frame-rotation charge conveys whether the two WPs can disappear from the $C_{2}\mathcal{T}$-plane.

Four scenarios are possible, as illustrated in Fig. 5a–b which displays the braiding of two principal WPs (marked in green vs. magenta to distinguish their chirality) around an adjacent node (gray). Starting with two WPs of equal chirality within the $C_{2}\mathcal{T}$-plane, they can either leave the plane (A) or bounce within the plane (B) when collided. Considering instead two principal WPs of opposite chirality, they can either annihilate (C) or bounce (D). In cases (B) and (D), the colliding WPs carry a nontrivial frame-rotation charge (i.e. Euler class), which obstructs their disappearance from the $C_{2}\mathcal{T}$-plane, irrespective of their chirality. In cases (A) and (C), the frame-rotation charge is trivial, allowing the pair of principal WPs to disappear from the plane.

Non-Abelian Weyl points with mirror symmetry

Additional crystalline symmetries may interact non-trivially with the non-Abelian band topology. Here, we consider the presence of a mirror symmetry $m$, which facilitates a mirror-invariant $m$-plane (cyan in Fig. 5d) perpendicular to the $C_{2}\mathcal{T}$-plane (orange in Fig. 5d). Recall that mirror symmetry can stabilize nodal-line degeneracies of bands with opposite mirror eigenvalues Fang:2016; Bzduvsek16nodal.

Let us consider a WP (green in Fig. 5c) stabilized inside $C_{2}\mathcal{T}$-plane on one side of the mirror. Such a WP has a mirror-related partner of opposite chirality (magenta in Fig. 5c) on opposite side of the $m$-plane. Moving the two WPs together locally inverts the two bands along the intersection ${m}\!\cap\!{C_{2}\mathcal{T}}$ of the two planes. We distinguish two possibilities. (1) If the two bands have the same mirror eigenvalue, they hybridize inside the $m$-plane, resulting in an avoided crossing, and the two WPs annihilate upon collision. (2) If the two bands have opposite mirror eigenvalue, they cannot hybridize. Their crossing is stable, implying that the two WPs convert to a NL Sun:2018. Note that the resulting NL still crosses the $C_{2}\mathcal{T}$-plane at two points (brown dots in Fig. 5c), which corresponds to the “bouncing” of two nodes inside the ${C_{2}\mathcal{T}}$-plane (white arrows in Fig. 5d) as described above as (D). Curiously, the fate of the reverse process, i.e. two NLs colliding inside $m\cap C_{2}\mathcal{T}$, does not uniquelly follow from the relative mirror eigenvalue of the principal bands. Nevertheless, we find that the outcome of such collisions can be predicted by computing the Euler class of the two NLs passing through the $C_{2}\mathcal{T}$-plane.

Node conversion in ZrTe

Zirconium telluride (ZrTe, space group #187) material class includes triple-point metals with a non-saturating quadratic magnetoresistance Zhu:2016; Weng:2016; He:2017; Ma:2018. Under ambient conditions, the band structure of ZrTe further exhibits 6 pairs of WPs (green and magenta in Fig. 6a) related by mirror ($m$) symmetry (cyan sheet in Fig. 6) near the K point of BZ, all of them lying inside vertical $C_{2}\mathcal{T}$-planes Weng:2016 (orange sheet in Fig. 6a). The WPs are located only $50\,\textrm{meV}$ above the Fermi level, which could possibly be further lowered by doping. These properties make ZrTe an ideal platform to study the interplay of the non-Abelian band topology with crystalline symmetry.

By performing high-precision first-principles computation including hybrid functional HSE06 vasp; wannier90; wanniertools (see Methods and Extended Data Figs. 2 and 3), we find that a $2\%$ compressive uniaxial strain in $z$-direction brings the pairs of WPs of ZrTe together at the $m$-plane. Although experimentally challenging, the relatively low Young modulus ($122\,\textrm{GPa}$ in the $z$-direction Guo:2017) may allow for angle-resolved photoemission spectroscopy (ARPES) of ZrTe under large strains. The relative mirror eigenvalue of the two bands forming the WPs are opposite (see Extended Data Fig. 2), hence the colliding WPs convert into NLs (brown lines in Fig. 6d). Further increasing the strain to $2.6\%$ fuses three elongated NLs into two concentric NL rings (Fig. 6e).

The reverse process, i.e. relaxing the strain of compressed ZrTe, exhibits collisions of two NLs within the $C_{2}\mathcal{T}$-plane along both the $\Gamma\textrm{K}$ and the KM high-symmetry lines. However, only the collisions along $\Gamma\textrm{K}$ eject pairs of WPs, while the collisions along KM do not. We compare the two situations by computing the Euler class (3) inside the red resp. the blue rectangular regions shown in Fig. 6e, each traversed by two nodal lines, using the algorithm outlined in Sec. H of SI. We find that the ejection of WPs from colliding NLs corresponds to a non-trivial Euler class on the rectangular region, cf. Fig. 6f and g, consistent with our theoretical predictions.

Further material candidates

We find that ZrTe is not the only WC-type material that supports WPs with a non-trivial Euler class. Through first-principles calculations, we confirm that WC-type MoP and NbS also exhibit WPs under ambient conditions. Among them, the WPs of MoP (plotted in Extended Data Fig. 4) were observed with ARPES Lv2017. We further find that there are NLs in ambient MoC and MoN. Those nodal lines can be converted into WPs by appropriate pressure. However, these four materials are not ideal candidates to validate our theory because (i) their Young modulus is larger than that of ZrTe, and (ii) the WPs and NLs are located about $1\,\textrm{eV}$ above or below the Fermi level. Notably, the pairs of WPs in the $C_{2}\mathcal{T}$-invariant plane of TaAs, which are experimentally well established Lv:2015; Xu:2015, also carry nontrivial Euler class (see Extended Data Figs. 5 and 6). Additional first-principles data for all the mentioned materials appear in Extended Data Fig. 7 and are discussed in Methods.

Other experimental realizations

While the presented data for the ZrTe material class underpin the predictive power of our analysis, we also mention other possible general directions. In particular, we anticipate that recent progress in the analysis of van-der-Waals heterostructures should entail an interesting research avenue. Studies into twisted bilayer graphene Cao_2018 have identified the existence of non-trivial band insulators in terms of the above characterization under $C_{2}\mathcal{T}$ symmetry Ashvin_2019; Ahn:2019; Bouhon2018_fragile; Song2019. The flexibility of the stacking direction (also under stress, strain and voltage/potential differences for e.g the $p_{z}$ band Ashvin_2019) could potentially be utilized to realize the reciprocal braiding phenomenon in such systems. Regarding explicitly three-dimensional Weyl semimetals, we remark the pioneering experimental efforts to move the WPs of $\textrm{WTe}_{2}$ through $\boldsymbol{k}$-space by driving an optical “shear” phonon mode Sie:2019.

Apart from the solid-state route, given the concrete nature of the braiding protocol presented in Fig. 2, another promising direction to access this physics entails implementation in cold-atom or photonic systems. Indeed, recent experimental studies have reported techniques to measure geometric concepts related to Euler class, such as the Wilson phases Li_2016. Furthermore, upon the influence of an external force, elements of the Berry-Wilczek-Zee connection Wilczek:1984 can be obtained, and the reconstruction of Berry curvature using tomography has been experimentally achieved Flaschner_2016. Similarly, photonic systems offer a platform to implement our scheme. Models with three or even four momentum dimensions are at present routinely implemented in topological photonics Lu:2013; Lu:2015; Zilberberg_2018, opening a possibility to emulate the interplay of the non-Abelian band topology with 3D point-group symmetry analogous to the ZrTe material class.

Conclusions

We presented an elementary protocol that elucidates a non-Abelian braiding effect for momentum-space band nodes, and we discussed three mathematically rigorous ways to characterize these phenomena based on frame rotations Wu:2018b, Eules class Ahn:2019, resp. Wilson loops Bouhon2018_fragile. We emphasize that the novel vantage point reached by considering different partitionings of bands via two energy gaps into three groups, namely (i) many occupied, (ii) two principal, and (iii) many unoccupied bands, opens up an avenue to new theoretical as well as experimental studies, as it allows for more intricate topological structures. Importantly, we considered the interplay of the non-Abelian topology with point-group symmetry in 3D, uncovering novel phase transitions of Weyl points in $C_{2}\mathcal{T}$-symmetric systems. The discussed models and phenomena are within current experimental reach, and the predictive power of our framework is reflected in specific material predictions.

Online content

The Methods section contains additional information on (i) the elementary braiding Hamiltonian, (ii) topology of complex and real vector bundles, (iii) the Wilson-flow algorithm, and (iv) the first-principles calculations. The Supplementary Information (SI) file contains information about (a) $C_{2}\mathcal{T}$-symmetric tight-binding models, (b) $\boldsymbol{k}$-local tight-binding braiding protocol, (c) the reality condition in $C_{2}\mathcal{T}$-symmetric models, (d) geometric interpretation of Euler form in three-band models, (e) behavior of Euler form at principal nodes, (f) Eq. (3) for manifolds with a boundary, (g) relation between Euler class and frame-rotation charge, and (h) the Euler-form integration algorithm. The Extended data contain one figure illustrating the Wilson-flow algorithm, and six figures with additional first-principles data for ZrTe, for other WC-type materials, and for TaAs.

References