Fig. 1. Non-Abelian topological charges and 2D band diagrams. (a) Non-Abelian topological charge +i; (b) non-Abelian topological charge +j, which can be expressed as +j=i⋅-k according to the quaternion group relations; (c) non-Abelian topological charge +k; (d) non-Abelian topological charge -1

Fig. 2. Non-Abelian topological edge states. (a) A model of one-dimensional three-band system; (b) edge states corresponding to non-Abelian topological charge +i; (c) edge states corresponding to non-Abelian topological charge +j; (d) edge states corresponding to non-Abelian topological charge +k; (e) edge states corresponding to non-Abelian topological charge -1

Fig. 3. Experimental demonstration of non-Abelian topological charges in a three-band transmission line network^[13]. (a) Transmission line network composed of cables; (b) band diagram obtained from theoretical predictions and experimental measurements, showing the validation of boundary states; (c) rotation of eigenstates as determined from theory and experiments; (d) non-Abelian quotient relation; (e) explanation of non-Abelian quotient relation

Fig. 4. Four-band non-Abelian topological charge^[14]. (a) Rotation on two orthogonal planes in four-dimensional space; (b) difference between ±q1234 on the Clifford torus; (c) simulation of -q1234 in the transmission line network and experimental verification of its edge states

Fig. 5. Non-Abelian topology in Floquet systems^[15]. (a) Two Hamiltonian models used to construct the driven system; (b) band gaps exist between all energy bands in the Floquet system; (c) band structure with topological charge +1; (d) spatial distribution of the energy bands under open boundary conditions when the topological charge is +1

Fig. 6. Topological constraints between nodal lines^[20]. (a)‒(c) Topological charge sign changes after adjacent band node loops thread through each other; (d) topological charge sign remains unchanged after non-adjacent band node loops thread through each other

Fig. 7. Realization of nodal link from triple-point^[21]. (a) (b) Band structures along high-symmetry lines with and without triple-point; (c) (d) nodal line structures before and after applying the rotational symmetry-breaking perturbation, and the green path γ0 carries a quaternion charge of -1

Fig. 8. Earring nodal link in photonic crystals^[22]. (a) Triple-point in the nodal line structure; (b) earring nodal link evolved from the triple-point, and a band gap inversion occurs from the left diagram to the right diagram; (c) photonic crystal unit cell; (d) earring nodal link in the momentum space of the photonic crystal, and a band gap inversion occurs from the left diagram to the right diagram

Fig. 9. Seemingly contradictory node-link structures in photonic metamaterials^[23]. (a) Orthogonally chained nodal rings between the 2nd and 3rd bands in the momentum space; (b) after considering the braiding with adjacent blue nodal lines, the change in the direction of the nodal lines is observed, and an inconsistency is found between the two time-reversal-symmetry-related π1 loops; (c) predicted structure of the purple nodal lines, which counteract the braiding effect induced by the blue nodal lines; (d) split-ring resonator structure; (e) nodal ring from the 2nd and 3rd bands is shown in red, and the predicted nodal lines from the 3rd and 4th bands are shown in purple

Fig. 10. Characterization of the source or sink of non-Abelian frame charge flow^[24]. (a) Intersection of two identical band gap nodal lines, and different loops carry different topological charges; (b) after adding adjacent band gap nodal lines, different loops carry the same topological charge of -1; (c) braiding can also be represented as a triple-point; (d) after the appearance of orange nodal line, the horizontal loop carries a topological charge of -1; (e) after the appearance of orange nodal line, the vertical loop carries a topological charge of -1; (f) as the perturbation tends to zero, the orange nodal line shrinks to infinitesimal and becomes the triply degenerate photonic Γ point

Fig. 11. Collision of three-band degeneracy points^[28]. (a) Two-dimensional band structure at t=0; (b) positions of the main node and adjacent nodes in the Brillouin zone, with real brown and imaginary purple lines representing two paths around the main node; (c) cumulative framework rotation angles calculated along the two paths in Fig.11 (b); (d) band structure before and after the nodal collision, with orange arrows indicating the direction of node movement as time increases

Fig. 12. Nodal braiding process in acoustic materials^[35-36]. (a) Experimental setup for measuring bulk and edge band structures; (b) acoustic band structure of the metamaterial, and the color represents the amplitude of measured acoustic signal after Fourier transformation (related to the acoustic band structure), while the white lines represent the computed acoustic band structure; (c) unit cell structure of another acoustic metamaterial; (d) band structure plotted along the diagonal of square Brillouin zone, with blue and red arrows indicating the direction of motion of the band nodes; (e)(f) measured spectra of acoustic metamaterial

Fig. 13. Evolution of non-Abelian topological charge nodes in the Kagome lattice with varying magnetic flux^[37]. (a)‒(c) Band structures (top panels) and node evolution in gaps I and Ⅱ (bottom panels) as the magnetic flux increases and the phase increases from φ=0 to φ=0.162π [In gap Ⅱ, solid (hollow) blue triangles represent Dirac nodes with quaternion frame charge Q=i-i; in gap I, solid (hollow) red pentagrams denote Dirac nodes with frame charge Q=k-k, solid (hollow) green quadrangular stars indicate linear triply-degenerate points with frame charge Q=j-j, and solid (hollow) brown circles correspond to quadratic nodes with frame charge Q=1-1]; (d) band structure of the system at φ=0.162π; (e) input at different energies corresponds to the energies of antichiral edge states in gap Ⅱ (top) and gap I (bottom), respectively

Fig. 14. Triple degeneracy^[46]. (a) Triple points (yellow points) formed by the intersection of two bands in a three-band model; (b) total flux is 0, corresponding to χ=0; (c) total flux is finite, leading to χ=2

Fig. 15. Phenomena of non-Abelian gauge fields^[55]. (a) (b) ZB effect induced by a synthetic non-Abelian magnetic field and by a synthetic non-Abelian electric field in a biaxial non-magnetic medium, respectively, where the upper panels are the full-wave simulated intensity distributions, and the lower panels show the intensity centroid trajectories with numerical (black circles) and analytical (red curves) results; (c) schematic of non-Abelian AB system; (d) corresponding intensity interference ψ²y and the two Euler angles α, β of the local spin sy on the screen, where the blue circles and red curves represent the simulated and theoretical results, respectively, the green curve corresponds to the case with gauge potential A=0, and δθ and b represent the phase shift and relative amplitude with respect to the case of A=0

Fig. 16. ZB effect induced by a non-Abelian electric field^[56]. (a) ZB effect in biaxial dielectric media at different rotation angles θ; (b) microwave metamaterials; (c)‒(e) ZB effect obtained at 12.675 GHz (top), 12.800 GHz (middle), and 12.950 GHz (bottom), respectively [(c) ZB effect intensity distribution; (d)(e) simulated and experimental results of the field distribution along black and bluish-green lines in Fig. 16 (c), respectively]