Abstract
Very recently, increasing attention has been focused on nonAbelian topological charges, e.g., the quaternion group Q_{8}. Different from Abelian topological band insulators, these systems involve multiple entangled bulk bandgaps and support nontrivial edge states that manifest the nonAbelian topological features. Furthermore, a system with an even or odd number of bands will exhibit a significant difference in nonAbelian topological classification. To date, there has been scant research investigating evenband nonAbelian topological insulators. Here, we both theoretically explore and experimentally realize a fourband PT (inversion and timereversal) symmetric system, where two new classes of topological charges as well as edge states are comprehensively studied. We illustrate their difference in the fourdimensional (4D) rotation sense on the stereographically projected Clifford tori. We show the evolution of the bulk topology by extending the 1D Hamiltonian onto a 2D plane and provide the accompanying edge state distributions following an analytical method. Our work presents an exhaustive study of fourband nonAbelian topological insulators and paves the way towards other evenband systems.
Similar content being viewed by others
Introduction
In mathematics, Abelian operators are commutative, meaning that the result of two successive operations does not depend on the order in which they are written. If we focus on a single bandgap, then topological physical systems^{1,2,3,4,5,6} are usually classified by Abelian groups, with the prime example being the tenfold classification^{7,8} of Hermitian topological insulators and superconductors. Once multiple bandgaps are collectively considered, their coupling introduces richer physics that can make the classification nonAbelian^{9,10,11,12,13}. A classic example is the quaternion group \({Q}_{8}=\left\{+1,\pm i,\pm j,\pm k,1\right\}\) with \({i}^{2}={j}^{2}={k}^{2}={ijk}=1\), which has been used to classify the topological line defects in biaxial nematic liquid crystals^{14}. Very recently, nonAbelian groups have been used to describe the admissible nodal line configurations^{12,15,16}, Dirac/Weyl point braiding^{13,17,18}, and intriguing triple nodal points^{19,20,21} in PT (inversion and timereversal) symmetric systems. When more bands are involved, richer nonAbelian topological charges emerge^{9}. Especially for systems with an even number of bands, several new classes of nonAbelian topological charges deserve special attention. A simple argument for this is that the evendimensional special orthogonal groups, i.e., \({SO}(2N)\), with \(N\) indicating a positive integer, contain inversion symmetry, i.e., \({I}_{2N}\) (the negative \(2N\times 2N\) identity matrix).
Results
NonAbelian topological charges in fourband models
Here, for simplicity, we focus on a fourband PT symmetric system. Choosing an appropriate basis, the Hamiltonian can take real forms, i.e., \(H\left(k\right)={H}^{\ast }(k)\). When simultaneously considering all three bandgaps between any two adjacent bands, the configuration space of the Hamiltonian is \({M}_{4}=O(4)/{{\mathbb{Z}}}_{2}^{4}\), with \(O(4)\) being the 4dimensional (4D) orthogonal group. This implies that the eigenstate frame remains intact under \(O(4)\) rotation, while \({{\mathbb{Z}}}_{2}^{4}\) indicates that each eigenstate has a gauge freedom of \(\pm 1\). The quantized charges that describe the underlying topology are found to be the nonAbelianbased homotopy group^{9} \({\pi }_{1}\left({M}_{4}\right)={Q}_{16}=\) \({{{{{{\rm{U}}}}}}}_{{n}_{i}\in \{0,1\}}\{\pm {e}_{1}^{{n}_{1}}{e}_{2}^{{n}_{2}}{e}_{3}^{{n}_{3}}\}\), where \({e}_{1},{e}_{2},{{{{{{\rm{and}}}}}}\,e}_{3}\) are the basis vectors of real Clifford algebra \(C{{{{{{\mathscr{l}}}}}}}_{{{{{\mathrm{0,3}}}}}}\) satisfying the relation \(\left\{{e}_{i},{e}_{j}\right\}=2{\delta }_{{ij}}\) (see Supplementary Note 1). There are 16 elements in the group and 10 conjugacy classes in total (see Table 1, as indicated by the curly braces). Group multiplication can be simply carried out using the above relation, i.e., \(\left({e}_{1}{e}_{2}\right)\left({e}_{1}{e}_{3}\right)={e}_{1}{e}_{1}{e}_{2}{e}_{3}={e}_{2}{e}_{3}\). Although the labels with the Clifford algebra basis (see the 1st column of Table 1) are convenient for group multiplication, decoding the underlying physical meaning is not straightforward. To relate the charges to rotations of the eigenstates, we rename all the charges onetoone, as shown in the 2nd column of Table 1. For example, we will see that \(\pm {q}_{12}\) indicate that both the 1st and 2nd bands acquire Zak phases of \(\pi\) due to the rotation of their respective eigenvectors. Figure 1a shows the representative elements and their multiplication relations, and the corresponding full multiplications are listed in Supplementary Tables 1 and 2. One may also note that the paths (arrows) bridging two elements are not unique. This means that the nonAbelian topological phase transitions are multiplepath transitions, which is different from the singlepath transitions in Abelian systems^{22}.
In the following, we study the topological properties of these charges. After topological band flattening, the mentioned PT symmetric fourband Hamiltonian can take the form of \(H(k)=R\left(k\right){I}_{1234}\left(k\right){R}^{T}\), with \(R\left(k\right)\in {SO}(4)\) being the 4D special orthogonal group, \(k\in [\pi ,\pi ]\) being the first Brillouin zone (FBZ) and \({I}_{1234}={diag}\left({{{{\mathrm{1,2,3,4}}}}}\right)\). The Hamiltonian has four real eigenvectors, as \(H(k)\leftn\right\rangle ={nn}\rangle\) with \(n=\)1, 2, 3, and 4. When \(k\) runs across the FBZ \((k=\pi \to \pi )\), rotation matrix \(R(k)\) continuously acts on eigenvector \({n}\rangle\), and one finally obtains \(+{or}{n}\rangle\) corresponding to a Zak phase of 0 or π, respectively. Without loss of generality, we assume \(R\left(k=\pi \right)={I}_{4}\). Because \({{{{{\rm{det }}}}}}(R)={\lambda }_{1}{\lambda }_{2}{\lambda }_{3}{\lambda }_{4}=1\), with \({\lambda }_{i}\) being the four eigenvalues of \(R(k)\), three exhaustive categories of possibilities at \(k=\pi\) can be easily found (see see Table 1) : (1) all four \({\lambda }_{i}=1\); (2) two \({\lambda }_{i}=1\), with the other two \({\lambda }_{i}=1\); and (3) all four \({\lambda }_{i}=1\).
The first category corresponds to two conjugacy classes \(\left\{+1\right\}\) and \(\left\{1\right\}\). Although they are indistinguishable from the Zak phase description, charge \(+1\) indicates that the trajectories of the eigenstate frame are contractible, while charge \(1\) indicates a noncontractible loop. Usually, charge \(1\) indicates that the eigenstate frame rotates by \(2\pi\) in a rotation plane (or topologically equivalent configurations)^{9,22}. We will see their difference more explicitly by extending the 1D Hamiltonian onto a 2D plane (Fig. 1d, e). The second category consists of six conjugacy classes that can be distinguished using singleband Zak phase arguments regarding which two of the four bands have Zak phases of \(\pi\). In the last category, all eigenstates flip their sign after \(k\) runs across the 1D FBZ. This category originates from the inversion symmetry \(({I}_{4})\) mentioned above. The two group elements (classes) also share the same Zak phase distribution and are indistinguishable from the conventional Abelian arguments. Their difference is reflected in the eigenstate rotation sense in four dimensions.
With setting \(\left(k\right)=\) \(\exp (\phi {\sum }_{i < j=1:4}{n}_{ij}{L}_{ij})\), we obtain the explicit form of the flatband Hamiltonian, where six skewsymmetric matrices \({L}_{{ij}}\) with entries \({\left({L}_{{ij}}\right)}_{a,b=1:4}={\delta }_{{ia}}{\delta }_{{jb}}+{\delta }_{{ib}}{\delta }_{{ja}}\) span the basis of Lie algebra \({\mathfrak{s}}{\mathfrak{o}}(4)\), \(\phi (k)\) is the rotation angle and \({n}_{{ij}}(k)\) determines the rotation plane. For example, the Hamiltonian of charge \({q}_{12}\) can be given with \(R\left(k\right)={{{{{\rm{exp }}}}}}\left(\frac{k+\pi }{2}{L}_{12}\right)\), while that of charge \(1\) can be obtained with \(R\left(k\right)={{{{{\rm{exp }}}}}}\left[\left(k+\pi \right){L}_{12}\right]\). Except for the charges \(\pm {q}_{1234}\), the rest have counterparts in the threeband systems^{22} studied previously. Thus, we mainly focus on the charges \(\pm {q}_{1234}\), which are unique in the fourband models.
While the nonAbelian topological charges are defined on 1D periodic lattices, their topological characteristics would be more straightforward to visualize after we generalize the 1D Hamiltonians onto a 2D extended plane, where each nonAbelian topological charge characterizing the 1D loop is reflected by the specific configuration of band degeneracies encircled by the 1D loop in the 2D plane. After trigonometrically expanding the Hamiltonian \(H(k)\), we make substitutions such as \({{{{{\rm{cos }}}}}}k\to \rho {{{{{\rm{cos }}}}}}k={k}_{1}\) and \({{{{{\rm{sin }}}}}}k\to \rho {{{{{\rm{sin }}}}}}k={k}_{2}\) and show the corresponding 2D bands in Fig. 1b–e. The original 1D Hamiltonian in \(k\) space is a unit circle (white circles in Fig. 1b–e) in the 2D extended plane that encircles nonremovable degeneracies explicitly exhibiting the underlying topological obstacles. For the charge \(+1\) (Fig. 1e), the topology is trivial, as there is no degeneracy enclosed by the white circles, while for charges \(\pm {q}_{{mn}}\) (Fig. 1b) and \(1\) (Fig. 1d), the 1D unit circles enclose linear and quadratic degeneracies, respectively. These 2D degeneracies topologically contribute to edge/domainwall states of the 1D systems; i.e., the linear/quadratic degeneracy implies one/two topologically protected edge states.
The charge \({q}_{1234}\) can be factorized as \({q}_{1234}={q}_{12}{q}_{34}\), \({q}_{1234}={q}_{14}{q}_{23}\), and \({q}_{1234}={q}_{13}{q}_{24}\) (the minus sign is induced by the odd permutation of subscripts). Note that the two factors in nodal links are commutative, i.e., \({q}_{12}{q}_{34}={q}_{34}{q}_{12}\), which means that all nodes formed by more distant (i.e., sharing no common band) pairs of bands commute^{9}. In Fig. 1c, we show the corresponding extended 2D band degeneracies of the three cases. They all belong to the same charge and can thus be continuously transformed into each other without closing the bandgap (see below). The charge \({q}_{1234}\) shares the same 2D band degeneracies with \({q}_{1234}\). Note that \(\pm {q}_{1234}\) belong to two different conjugacy classes, which is one of the key points that fundamentally distinguishes them from the charges \(\pm {q}_{{mn}}\). We will show their topological differences in the following section from the eigenstate rotation perspective.
We note that the nodal ring degeneracies in Fig. 1b (\(\pm {q}_{14}\)) and c (\({q}_{13}{q}_{24}\)) are accidental in the flatband models, and each will be split into linear Dirac cones in more general situations (see below). Other triple degeneracies are similar to charges \(\pm j\) in threeband models^{22}, where three bands are involved. The fourfold degeneracy in Fig. 1c (\({q}_{14}{q}_{23}\)) is also admissible rather than stable here.
Eigenstates on the threesphere \({S}^{3}\)
Here, we illustrate rotation configurations pertaining to different charges of the generalized quaternion group \({Q}_{16}\). The normalized eigenstates of \(H(k)\) are all real and can be parametrized by Hopf coordinates \((\alpha ,\eta ,\beta )\) on the threesphere  \({S}^{3}\). Their four components can be written as \(\left(u={{{{{\rm{cos }}}}}}\alpha {{{{{\rm{sin }}}}}}\eta ,x={{{{{\rm{sin }}}}}}\alpha {{{{{\rm{sin }}}}}}\eta ,y={{{{{\rm{cos }}}}}}\beta {{{{{\rm{cos }}}}}}\eta ,z={{{{{\rm{sin }}}}}}\beta {{{{{\rm{cos }}}}}}\eta \right)\), where \(\alpha\) and \(\beta\) correspond to the two rotation angles in the two orthogonal invariant planes, as shown in Fig. 2a (also see Supplementary Note 2: Rotations in four dimensions^{23,24,25}), while \(\eta\) determines the proportions projected onto the two planes. When \(\alpha \ne 0\) and \(\beta =0\) (or \(\alpha =0\) and \(\beta \ne 0\)), the rotations are called single rotations. For example, all ideal rotations \(R(k)\) with \(k=\pi \to \pi\) enabling charge \({q}_{12}\) belong to the case with the settings \(\eta =\frac{\pi }{2}\) and \(\alpha =\frac{k+\pi }{2}\), where “ideal” indicates the flatband model given above. Note that all general models can be continuously transformed into the ideal flatband model, and they are topologically equivalent. Other charges, including \(\pm {q}_{{mn}}\) and \(1\), can be realized in a similar manner. Clearly, the eigenstates in one plane (i.e., \({oyz}\) plane when \(\eta =\frac{\pi }{2}\)) can be fixed for these cases, while they rotate on the other orthogonal plane (i.e., \({oux}\) plane). In other words, the ideal rotations can be carried out in a 2D subspace. Notably, in contrast to on which plane the eigenstates rotate, the crucial property of these topological charges is that the eigenstate trajectories cannot contract to isolated points. The difference in charges \(\pm {q}_{{mn}}\) is reflected by which two bands (the mth and nth) are noncontractible, while charge \(1\) requires that all four trajectories cannot contract simultaneously.
When both \(\alpha \ne 0\) and \(\beta \ne 0\), the rotations are dubbed double rotations (Fig. 2a), where there are two possibilities: rotating on the two planes in the same (\(\alpha \beta > 0\)) or opposite \((\alpha \beta < 0)\) sense. The charges \(\pm {q}_{1234}\) have to be realized with continuous double rotations, which means that \(R(k)\) at each \(k\) point is a double rotation. Interestingly, when \(\eta =\frac{\pi }{4}\), the parametric set \((u,x,y,z)\) constructs a Clifford torus^{26}, which is the Cartesian product of two circles in \({{\mathbb{R}}}^{4}\) (e.g., \({S}_{A}^{1}\in {ou}x,{S}_{B}^{1}\in {oyz}{\,}{{{\rm{and}}}}{\,}{S}_{A}^{1}\times {S}_{B}^{1}\in {{\mathbb{R}}}^{4}\)). The Clifford torus can be stereographically projected^{26} into \({{\mathbb{R}}}^{3}\) as a conventional torus, i.e., \(\left(\frac{x}{1u},\frac{y}{1u},\frac{z}{1u}\right)\), on which we can pictorially illustrate the difference between charges \(\pm {q}_{1234}\) in the rotation sense of eigenstate trajectories, as shown in Fig. 2b. The two panels correspond to \(\alpha =\beta =\frac{k+\pi }{2}\) (left, \({q}_{1234}\)) and \(\alpha =\beta =\frac{k+\pi }{2}\) (right, \({q}_{1234}\)).
We further propose another orthographic projection method, which projects each 4D trajectory into 3D space from four orthogonal views. This is similar to the threeview drawing, which is the orthographic projection from 3D space to 2D plane. Taking the first panel of Fig. 2c as an example, we plot the trajectories in the \({xyz}\) subspace to obtain an orthographic projection from the view of the \(u\) direction. Figure 2c, d correspond to \(+{q}_{1234}\) and \({q}_{1234}\), respectively, where eigenstate trajectories are mapped onto four solid spheres in \({{\mathbb{R}}}^{3}\). One can see that their main difference is that the rotation directions in the \({oux}\) plane are opposite. Orthographic projections for other charges are listed in Supplementary Figs. 1–4. In Fig. 2e, we show the topological phase transition between them, where there are inevitably two linear crossings between the first and second bands as system parameter \({w}_{{AB}}\) changes (without relying on a joint basepoint, as they belong to different classes).
Zak phases and evolution of edge states
After understanding the nonAbelian topological charges from the perspective of eigenstate frame rotations, we now show their relations to the Zak phases of each band as well as edge/domainwall states. In a PTsymmetric system, the Zak phases of each band take a quantized value of \(0\) or \(\pi\), and the values are shown in Table 1; i.e., \({\lambda }_{i}=1\) indicates a Zak phase of \(\pi\). We further refine the Zak phase of \(\pi\) to be \(\pm \pi\), where “\(\pm\)” is used to differentiate between charges \(\pm {q}_{{mn}}\) (two elements in the same conjugacy class). All of the corresponding singleband Zak phases are exhaustively summarized in Fig. 3a. For charges \(\pm {q}_{{mn}}\), two corresponding bands with noncontractible eigenstate trajectories carry Zak phases of \(\pm \pi\), and the bandgap sandwiched by them supports edge states at hard boundaries of a finite lattice. We take the case of \(\pm {q}_{12}\) as an example, as shown in Fig. 3b. The edge states of other \(\pm {q}_{{mn}}\) charges are shown in Supplementary Fig. 6. We label charge \(1\) with \(2\pi\), which indicates noncontractible \(2\pi\) rotation here^{22}.
For charges \(\pm {q}_{1234}\), two eigenstates rotate by \(\pi\), while the other two rotate by \(\pm \pi\) when \(k=\pi \to \pi\). As shown in Fig. 1c, there are three ways of factorization. We further schematically show them in Fig. 3c, where each doubleheaded arrow represents one factorization. The commutative property between two factor charges, i.e., \({q}_{12}{q}_{34}={q}_{34}{q}_{12}\), is implied by the doubleheaded arrows. The fact that \({q}_{12}{q}_{34}\) (typeI), \({q}_{13}{q}_{24}\) (typeII) and \({q}_{14}{q}_{23}\) (typeIII) are the same element in the group can be visualized by constructing a transformation between them without gap closing. The continuous transition between different factorizations can be explicitly parameterized. For example, from \({q}_{12}{q}_{34}\to {q}_{13}{q}_{24}\), we have \(H\left(k\right)={R}_{2}{R}_{1}{I}_{1234}{R}_{1}^{1}{R}_{2}^{1}\), with \({R}_{1}\left(k\right)={{{{{\rm{exp }}}}}}\left[\left(k+\pi \right)/2\left({{{{{\rm{cos }}}}}}{\theta }_{I\to {II}}{L}_{12}{{{{{\rm{sin }}}}}}{\theta }_{I\to {II}}{L}_{13}\right)\right]\) and \({R}_{2}(k)= {{{{{\rm{exp }}}}}}\left[\left(k+\pi \right)/2\left({{{{{\rm{cos }}}}}}{\theta }_{I\to {II}}{L}_{34}+{{{{{\rm{sin }}}}}}{\theta }_{I\to {II}}{L}_{24}\right)\right]\), as shown in Fig. 3c (see the evolution of eigenstate trajectories in Supplementary Fig. 3). In other words, the pair of two orthogonal invariant planes rotates with \({\theta }_{I\to {II}}\). We further study the accompanying evolution of edge states at hard boundaries, as shown in Fig. 3d–f. The analytical results are \({E}^{\pm }=\frac{5}{2}\pm \frac{\sqrt{2}}{4}\sqrt{5+3{{{{{\rm{cos }}}}}}2{\theta }_{I\to {II}}}\), \({E}^{\pm }=\frac{5}{2}\pm \frac{1}{2}{{{{{\rm{cos }}}}}}{\theta }_{{II}\to {III}}\), and \({E}^{\pm }=\frac{5}{2}\pm {{{{{\rm{sin }}}}}}{\theta }_{{III}\to I}\). Detailed analytical methods are provided in Supplementary Note 3 and 4. There are a total of two edge states pumping between different bandgaps. Their field distributions are given in Supplementary Figs. 7–9. The existence of these edge modes can be heuristically inferred by examining the band degeneracies of the extended 2D model. In Fig. 3g–i, we show the radial cuts of their extended 2D bands, where one can easily find that each linear degeneracy point at \({k}_{r}=0\) implies the position of each edge state in Fig. 3d–f, respectively. Note that in these flatband cases, only the degeneracies at \({k}_{r}=0\) imply topological edge states, while other degeneracies \(({k}_{r}\ne 0)\) accidentally emerge from the 2D nodal rings (e.g., see Fig. 1c), which have no topological implication.
We also show the edge state evolution for charge \(1\) in Supplementary Figs. 10 and 11 (see the analytical solutions in Supplementary Note 4). Along the 12 edges of the charge \(1\) octahedron (Supplementary Fig. 10a), the evolution shows strong resemblance to the threeband models^{22}. This occurs because only three bands participate in the edge state pumping. As such, all these transitions can be understood via the rotations of eigenstates in the subgroup \({SO}(3)\), while the fourth band is fully fixed and decoupled. One other important note is that there are 12 possible routes rather than 15 (naively from \(C\left(6,2\right)=15\)) because direct evolution between two orthogonal planes (or between the diagonal points linked by the dashed lines in Supplementary Fig. 10a, b), e.g., between \({q}_{12}^{2}\) and \({q}_{34}^{2}\), is impossible. We also find that the transition can take arbitrary routes on the 8 faces of the charge \(1\) octahedron (see an example in Supplementary Figs. 10d and 11). Supplementary Fig. 12 shows the evolution of 2D extended band degeneracies, which help us understand the pumping of edge states accordingly, e.g., the double quadratic or triple linear degeneracies at \({k}_{r}=0\) predict the emergence of topological edge states^{22}.
Observation of charges \(\pm {q}_{1234}\) in a transmission line network
To realize and characterize charges \({\pm q}_{1234}\), we designed a transmission line network^{22,27,28} (see the sample photo in Supplementary Fig. 13) consisting of 11 unit cells. There are four metaatoms A, B, C, and D in one unit cell. The realspace Hamiltonian reads (see details in Supplementary Note 3)
where \({c}_{X,n}^{{{\dagger}} }\) and \({c}_{X,n}\) are creation and annihilation operators on the sublattice ‘\(X\)/\(Y\)’ and site ‘\(n\)’, respectively. To realize an explicitly real Hamiltonian in momentum space, we introduce imaginary hoppings^{22}. More details on the experimental realization are provided in Methods section and Guo et al.^{22}.
The left two panels (BulkS) in Fig. 4a show the numerically calculated and experimentally measured energy bands. We plot the corresponding eigenstate trajectories of the four bands in Fig. 4c, d. In the experimental model, at each \(k\), we can see that one rotation plane is spanned by the eigenvectors of the first and second bands and the other rotation plane is spanned by those of the third and fourth bands. We can expect two topological edge states in total: one is located in the first bandgap (sandwiched by the first and second bands), while the other is located in the third bandgap (formed by the third and fourth bands). The rightmost panels (EdgeS) of Fig. 4a confirm our expectation. The detailed field distribution is provided in Supplementary Fig. 14b. The distribution of edge states can also be directly inferred from the 2D extended energy bands, as shown in Fig. 4b, where there is one linear Dirac cone between the first/third and second/fourth bands.
Domainwall states between charges \(+{q}_{1234}\) and\({q}_{1234}\)
If two samples with different nonAbelian topological charges meet at a domain wall^{22}, then some domainwall states (DWSs) will emerge, and their existence can be predicted by defining a “domainwall charge” \(\Delta Q={Q}_{L}/{Q}_{R}\). Here, \({Q}_{L}\) and \({Q}_{R}\) are the nonAbelian topological charges of the left and right samples, respectively. The quotient charge \(\Delta Q\) is also an element of the nonAbelian group and governs the properties (including both location and number) of the DWSs. We note that the appearance of the domainwall charge −1 in the threeband system can only be well defined by assuming a joint \(k\)space basepoint between the left and right samples^{22}. Otherwise, one cannot distinguish two nonAbelian topological charges (e.g., \(+i\) and \(i\)) in the same conjugacy class of the threeband system, and thus, the domainwall charge becomes illdefined. In the fourband system, however, there exists a basepointfree domainwall charge taking a value of \(1\) between charges \(+{q}_{1234}\) and \({q}_{1234}\). This occurs because they belong to two different conjugacy classes.
In the experiment, we construct a domain wall (blue spheres in Fig. 4e) between charges \(\pm {q}_{1234}\), as shown in Fig. 4e, where we flip the directions of imaginary hoppings between metaatoms C and D, as denoted by the blue arrows, to realize the charge \(+{q}_{1234}\) on the righthand side of the domain wall. Figure 4f shows the DWSs between them, where the left inset is the simulated energy levels and the right two insets indicate the measured spectra on the domain wall for two different excitation/probe locations accordingly. These results indicate that there are two nearly degenerate topological DWSs in the third bandgap. This is the same as the hard boundary edge states of charge \(1\) and thus confirms our prediction. The detailed field distribution is provided in Supplementary Fig. 15.
Observation of charges \(\pm {q}_{14}\) in a transmission line network
In addition, we experimentally studied charges \(\pm {q}_{14}\), which are also interesting in the fourband models, as they exhibit three edge states in the three bandgaps. As shown in Fig. 5, from the bulk bands (Fig. 5aBulkS), edge state distributions (Fig. 5aEdgeS) and eigenstate trajectories (Fig. 5c, d), the numerical calculations correctly predict the experimental results. Different from Fig. 1b (\(\pm {q}_{14}\)) of the flatband model, the 2D extended energy bands in Fig. 5b are bridged by three linear Dirac cones. As mentioned above, each implies one edge state (per edge), as verified in Fig. 5aEdgeS. For charges \(\pm {q}_{14}\), there is no complete bandgap in the 2D extended bands. They can be regarded as the generalization of charges \(\pm j\) in threeband models^{9,22}.
Discussion
Other general configurations of charges \(\pm {q}_{1234}\) are shown in Supplementary Figs. 16 and 17, corresponding to the factorizations of \({q}_{13}{q}_{24}\) and \({q}_{14}{q}_{23}\), respectively. As mentioned above, the ring degeneracy formed by the second and third bands in Fig. 1c (\({q}_{13}{q}_{24}\)) splits into two Dirac cones, as shown in Supplementary Fig. 16c, which further implies two edge states (per edge) in Supplementary Fig. 16d. The general model of charge \(1\) in Supplementary Fig. 18 shows one triple linear degeneracy, which is similar to what we have observed in the threeband models, such as the edge state distributions^{22}.
In the classic context, the transmission line network is extremely versatile and can be used to realize various lattice models, including pathdependent annihilation of Dirac points in 2D, braiding of Weyl points in 3D^{13}, accompanying dynamics of wavepacket propagation, etc. In the quantum context, all of the singleparticle topological phenomena can be well transferred. Furthermore, with introducing extra interaction and correlation physics, much more exotic topological braiding features are expected. Currently, the nonAbelian topological charges are limited to PT symmetry, it is very desirable to extend to other symmetry protections as well as nonHermitian systems^{29}.
Our exhaustive study of all nonAbelian topological charges of PT symmetric fourband Hamiltonians will constructively stimulate related research on 2D twisted bilayer graphene^{10,30,31}. The PT symmetric system also contributes to exotic fragile topological states^{32} and even topological effective gravitational theory^{33}. The studies can be easily transferred to other artificial platforms, including optical lattices^{34}, photonics^{35,36,37}, and phononics^{38}.
Methods
Experimental measurements
There are four metaatoms A, B, C, and D in one unitcell. The hoppings between two metaatoms are realized by connecting 2mlong coaxial cables (model: RG58C/U). To achieve the complex hoppings, we create a hidden dimension by placing four nodes in each metaatom so that four subspaces are allowed. Due to periodic connections in this hidden dimension, the four subspaces correspond to four pseudo angular momenta that are \({{{{{\rm{exp }}}}}}(i4{\varphi }_{n})=1\), with \({\varphi }_{1}=0,\;{\varphi }_{2}=\frac{\pi }{2},\;{\varphi }_{3}=\pi\; {{{{{{\mathrm{and}}}}}}}\;{\varphi }_{4}=\pi /2\). Through the specific excitation from a 4channel signal generator (Keysight M3201A), we carried out our experiments in the \({\varphi }_{2}=\pi /2\) subspace. The amplitude and phase of voltage of each metaatom are probed by an oscilloscope (Keysight DSOX2002A). After subsequent Fourier transformation, we obtain the energy bands and eigenstates in the momentum space. Supplementary Fig. 13 shows the specific transmission line network corresponding to charges \(\pm {q}_{14}\).
Data availability
The experimental data that support the findings of this study are available in DataSpace@HKUST with the identifier “https://doi.org/10.14711/dataset/VNMSFX”^{39}.
References
Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).
Qi, X.L. & Zhang, S.C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).
Ando, Y. & Fu, L. Topological crystalline insulators and topological superconductors: from concepts to materials. Annu. Rev. Condens. Matter Phys. 6, 361–381 (2015).
Chiu, C.K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016).
Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in threedimensional solids. Rev. Mod. Phys. 90, 015001 (2018).
Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019).
Schnyder, A. P., Ryu, S., Furusaki, A. & Ludwig, A. W. W. Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008).
Kitaev, A. Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22–30 (2009).
Wu, Q., Soluyanov, A. A. & Bzdušek, T. NonAbelian band topology in noninteracting metals. Science 365, 1273 (2019).
Ahn, J., Park, S. & Yang, B.J. Failure of NielsenNinomiya theorem and fragile topology in twodimensional systems with spacetime inversion symmetry: application to twisted bilayer graphene at magic angle. Phys. Rev. X 9, 021013 (2019).
Tiwari, A. & Bzdušek, T. NonAbelian topology of nodalline rings in PTsymmetric systems. Phys. Rev. B 101, 195130 (2020).
Yang, E. et al. Observation of nonabelian nodal links in photonics. Phys. Rev. Lett. 125, 033901 (2020).
Bouhon, A. et al. NonAbelian reciprocal braiding of Weyl points and its manifestation in ZrTe. Nat. Phys. 16, 1137–1143 (2020).
Mermin, N. D. The topological theory of defects in ordered media. Rev. Mod. Phys. 51, 591–648 (1979).
Wang, D. et al. Intrinsic inplane nodal chain and generalized quaternion charge protected nodal link in photonics. Light. Sci. Appl. 10, 83 (2021).
Park, H., Wong, S., Zhang, X. & Oh, S. S. NonAbelian Charged Nodal Links In Dielectric Photonic Crystal. arXiv:2102.12546 [physics.optics] (2021).
Peng, B., Bouhon, A., Monserrat, B. & Slager, R.J. NonAbelian Braiding Of Phonons In Layered Silicates. arXiv:2105.08733 [condmat.meshall] (2021).
Jiang, B. et al. Observation Of Nonabelian Topological Semimetals And Their Phase Transitions. arXiv:2104.13397 [condmat.meshall] (2021).
Lenggenhager, P. M., Liu, X., Neupert, T. & Bzdušek, T. Universal Higherorder Bulkboundary Correspondence Of Triple Nodal Points. arXiv:2104.11254 [condmat.meshall] (2021).
Guo, Q. et al. Observation of threedimensional photonic dirac points and spinpolarized surface arcs. Phys. Rev. Lett. 122, 203903 (2019).
Lenggenhager, P. M., Liu, X., Tsirkin, S. S., Neupert, T. & Bzdušek, T. From triplepoint materials to multiband nodal links. Phys. Rev. B 103, L121101 (2021).
Guo, Q. et al. Experimental observation of nonAbelian topological charges and edge states. Nature 594, 195–200 (2021).
PerezGracia, A. & Thomas, F. On Cayley’s factorization of 4D rotations and applications. Adv. Appl. Clifford Algebras 27, 523–538 (2017).
Le Bihan, N. The geometry of proper quaternion random variables. Signal Process. 138, 106–116 (2017).
Erdoğdu, M. & Özdemir, M. Simple, double and isoclinic rotations with a viable algorithm. Math. Sci. Appl. ENotes 8, 11–24 (2020).
McCuan, J. & Spietz, L. Rotations of the threesphere and symmetry of the Clifford torus. arXiv:math/9810023 [math.MG] (1998).
Zhao, E. Topological circuits of inductors and capacitors. Ann. Phys. 399, 289–313 (2018).
Jiang, T. et al. Experimental demonstration of angular momentumdependent topological transport using a transmission line network. Nat. Commun. 10, 434 (2019).
Ezawa, M. NonHermitian nonAbelian topological insulators with PT symmetry. Phys. Rev. Res. 3, 043006 (2021).
Song, Z.D., Lian, B., Regnault, N. & Bernevig, B. A. Twisted bilayer graphene. II. Stable symmetry anomaly. Phys. Rev. B 103, 205412 (2021).
Kang, J. & Vafek, O. NonAbelian Dirac node braiding and neardegeneracy of correlated phases at odd integer filling in magicangle twisted bilayer graphene. Phys. Rev. B 102, 035161 (2020).
Bouhon, A., Bzdušek, T. & Slager, R.J. Geometric approach to fragile topology beyond symmetry indicators. Phys. Rev. B 102, 115135 (2020).
Lopes, P. Le. S., Teo, J. C. Y. & Ryu, S. Effective response theory for zeroenergy Majorana bound states in three spatial dimensions. Phys. Rev. B 91, 184111 (2015).
Ünal, F. N., Bouhon, A. & Slager, R.J. Topological euler class as a dynamical observable in optical lattices. Phys. Rev. Lett. 125, 053601 (2020).
Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013).
Hafezi, M., Mittal, S., Fan, J., Migdall, A. & Taylor, J. M. Imaging topological edge states in silicon photonics. Nat. Photon. 7, 1001–1005 (2013).
Dutt, A. et al. A single photonic cavity with two independent physical synthetic dimensions. Science 367, 59 (2020).
Ma, G., Xiao, M. & Chan, C. T. Topological phases in acoustic and mechanical systems. Nat. Rev. Phys. 1, 281–294 (2019).
Jiang, T. et al. Replication Data For: Fourband Nonabelian Topological Insulator And Its Experimental Realization. https://doi.org/10.14711/dataset/VNMSFX (2021).
Acknowledgements
This work is supported by the Hong Kong RGC (AoE/P502/20, 16310420, and 16307821), the Hong Kong Scholars Program (XJ2019007), the KAUST CRG grant (KAUST20SC01) and the Croucher foundation (CAS20SC01).
Author information
Authors and Affiliations
Contributions
T.J., Q.G., B.Y. and C.T.C. conceived the idea; T.J. designed the transmission line network with input from Z.Q.Z. and C.T.C.; T.J. and Q.G. carried out all measurements; T.J., Q.G., R.Y.Z., Z.Q.Z., B.Y. and C.T.C. developed and carried out the theoretical analysis; B.Y. and C.T.C. supervised the whole project. T.J., Q.G. and B.Y. wrote the manuscript and the Supplementary Information with input from all other authors.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Peer review information Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Jiang, T., Guo, Q., Zhang, RY. et al. Fourband nonAbelian topological insulator and its experimental realization. Nat Commun 12, 6471 (2021). https://doi.org/10.1038/s41467021267631
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467021267631
This article is cited by

A second wave of topological phenomena in photonics and acoustics
Nature (2023)

Minimal nonabelian nodal braiding in ideal metamaterials
Nature Communications (2023)

Floquet nonAbelian topological insulator and multifold bulkedge correspondence
Nature Communications (2023)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.