Photonic non-Abelian topological insulators with six bands

Domain: physics
Published: 2026-02-21
Editorial outcome: minor_revision

Abstract
Non-Abelian topological insulators are multi-band systems, satisfying parity-time symmetry, characterized by fully gapped bands and whose topological charges are non-Abelian quaternions. These systems provide a richer landscape of edge and domain-wall states compared to that predicted by the Zak phase framework for Abelian topological insulators. Although non-Abelian topological insulators have been demonstrated in some proof-of-concept platforms, their realization in a photonic platform, where their characteristics are particularly valuable for on-chip applications, remains out of reach due to the lack of a feasible model. Here, we propose a minimal model for non-Abelian topological insulators and experimentally realize a six-band system at optical frequencies by using photonic waveguide arrays. A number of non-Abelian topological charges is experimentally demonstrated by probing the adiabatic evolution of edge states in a system with varying structural parameters. At the boundary between two samples with different non-Abelian topological charges, we experimentally observe domain-wall states that adhere to a non-Abelian quotient relation. The proposed idea can be generalized to an arbitrary number of bands and offer the flexibility to control multiple edge and domain-wall states in photonic devices.

Introduction
Research on topological insulators has advanced significantly in recent decades1–5. The Zak phase is a key concept in topological insulators which represents the Wannier center’s displacement and relates to polarization, leading to charge accumulation at sample boundaries6–9. It can be used to predict edge states and is quantized to either 0 or π in parity-time (PT) symmetric systems. In conventional bulk-boundary theory, the Zak phase difference Δv across a boundary (either 0 or π) of a band gap predicts the number of boundary modes, given by Δv=π. However, this criterion is inadequate in multi-band systems where the number of bands n≥310,11. In this case, it is useful to use non-Abelian band topology theory12–42 to understand the relationship between topological invariants and edge states in multi-band PT-symmetric systems. Unlike conventional band theory that focuses on single band or band gaps, the non-Abelian band topology theory considers the eigenstates of all bands globally, drawing analogy of their eigenframe rotation in momentum space to the rotation of liquid crystal molecules around a loop encircling nematic liquid crystal disclinations, with topological charges represented by quaternions (n=3) or generalized quaternions (n>3). Here, PT symmetry constrains the Hamiltonian to remain real across the Brillouin zone, which facilitates topological characterization within a Clifford algebra framework. Currently, this theory has been applied in some multi-band systems, including those with band degeneracy and those with fully gapped bands. The former, known as non-Abelian semimetal, is primarily used to investigate trajectory-dependent nodal point collisions13,19, admissible nodal line configurations14,16,20–24, and the Euler class26–29,36,37. The latter, known as non-Abelian topological insulator, has unconventional bulk-boundary correspondence, which extends beyond the conventional description based on integers18,38. In contrast to conventional theories where domain-wall states are determined by the difference in topological charges, non-Abelian topology theory predicts them based on the quotient of charges which are matrix-like entities18. Up to now, non-Abelian band topological systems have been experimentally realized in various platforms including metamaterials17,20–22, transmission line networks18,38 and acoustic systems19,24,40. Advancing the field into technological areas, such as photonics, is of great importance, as the non-Abelian features and their associated matrix-form operations are useful for various applications, including optical computing and topological quantum computing. However, a model for realizing non-Abelian band topological systems at optical frequencies remains elusive, and corresponding experimental demonstrations have yet to be achieved.

In this study, we propose a minimal model for non-Abelian topological insulators with time-reversal symmetry and experimentally realize a six-band system at optical frequencies. We show that the non-Abelian topological charges in the six-band system describe eigenvector frame rotations as Bloch momentum changes and are characterized by generalized quaternions, including ±q123456, which have no counterparts in systems with n<6 and exhibit unique edge/domain-wall state distributions beyond the single-band Zak phase prediction. Subsequently, we experimentally realize the proposed model at optical frequencies in a laser-written photonic waveguide array. We design and implement an experimental configuration in which light could encounter structures with different non-Abelian topological phases along the propagation direction. In this way, different non-Abelian topological charges can be revealed in experiments. The non-Abelian quotient relation is also experimentally substantiated by constructing an interface between two samples with different non-Abelian topological charges.

Results
Minimal non-Abelian topological insulator model with six bands
We first consider a quasi-1D tight-binding model as shown in Fig. 1a, which contains three coupled SSH chains. Each unit cell (dashed rectangle) contains six sites labeled “1” to “6”. The onsite energies and hopping strengths are chosen to ensure PT symmetry. The Hamiltonian is expressed as:

[Equation 1]

Here, h12 = w1 + v1 exp(-ik), h34 = w2 + v2 exp(-ik) and h56 = w3 + v3 exp(-ik), where k is the Bloch vector.

Before discussing the non-Abelian band topology, we first examine the correspondence between the single-band Zak phase and the existence of edge states. In Fig. 1b, c, we show the band structures (left panels) and energy spectra of finite samples with 20 unit cells (right panels) for two different parameter sets (see Supplementary Note 1). All the coupling strengths are set to be positive here. The Zak phases are labeled in the band structures. In the left panel of Fig. 1b, the Zak phases for the individual bands from first to sixth are all π, so if we sum up the Zak phases for the bands below a gap, the Zak phases for the first, third, and fifth gaps are π (dark gray), and the second and fourth gaps are 0 (light gray). According to the conventional bulk-boundary correspondence, the number of edge states in the first, third, and fifth gaps should be odd, while they should be zero in the second and fourth gaps. However, in the right panel of Fig. 1b, we find only three pairs of edge states in the third gap and none in the first and fifth gaps, which disagrees with the prediction from the Zak phase perspective. A similar analysis in Fig. 1c also shows deviations from conventional theory. Before explaining the failure of the Zak phase in predicting edge states, we first explain why this model can be considered a non-Abelian topological insulator.

According to the non-Abelian band topology theory, the PT-symmetric Hamiltonian H(k) can always be transformed into a real Hamiltonian He(k) through a unitary transformation He(k) = V*H(k)*Vy. The unitary matrix V can be derived from Takagi’s decomposition U = VVT, where U is a symmetric matrix representing the parity inversion operation. In our model, the unitary matrix V is chosen as [Equation 2].

After the unitary transformation, the obtained Hamiltonian is [Equation 3].

Here, he1 = w1 + v1 cos(k), he2 = w2 + v2 cos(k), he3 = w3 + v3 cos(k), he11 = v1 sin(k), he12 = v1 sin(k) and he33 = v2 sin(k). In this new Hamiltonian, all matrix elements are real, so the eigenvectors are real vectors denoted as u1, u2, u3, u4, u5, u6 T. These six real components define a 6D parameter space, and the evolution of the eigenvectors maps to trajectories within this parameter space as Bloch k varies. Considering all the band gaps between adjacent bands, the order-parameter space of the Hamiltonian is M = O(6)/O(1)6, where O(6) is the 6D orthogonal group and O(1)6 ≃ Z26 arises from the ±1 gauge freedom of the eigenvectors. The corresponding fundamental homotopy group is π1(M) = P6 ≃ Z25, which is formed by products of the ±1 gauge freedom, and the elements are [Equation 4].

The topological phase classification is determined by this non-Abelian fundamental group P6, and the group elements are referred to as the non-Abelian topological charges characterizing the topological phases. The group P6 has 64 elements grouped into 34 conjugacy classes: {+1}, {-1}, {±qij}, {±qijkl}, +q123456, and -q123456. The class {+1} denotes trivial rotation, and {-1} refers to a 2π rotation in any rotation plane. The ±qij represent ±π rotations in the rotation plane spanned by the eigenvectors of the i-th and j-th bands. The ±qijkl correspond to isoclinic double rotations of angle π in the 4D subspace of the i-th, j-th, k-th and l-th bands. The +q123456 and -q123456 correspond to triple rotations in the 6D space with opposite chirality. Figure 1d illustrates part of the group multiplication structure, showing elements as colored spheres and their multiplications with colored arrows. More group multiplication relationships are shown in Supplementary Tables 1-8.

Based on the above theoretical framework, the non-Abelian topological charges of the Hamiltonians in Fig. 1b, c are calculated to be q123456 and q1256, respectively (see Supplementary Note 2 for the calculation details). Their corresponding eigenvector trajectories within the u1u2u3, u3u4u6 and u4u5u6 subspaces are plotted in Fig. 1e, f, respectively, where the red, green, blue, purple, cyan, and yellow trajectories represent the tips of the eigenstates of the bands from first to sixth, with increasing line thickness showing k from -π to π. From Fig. 1e, all band eigenstates show a π rotation as the Hamiltonian evolves across 1D Brillouin zone, consistent with the charge q123456. In Fig. 1f, the first, second, fifth, and sixth bands rotate by π, while the third and fourth bands show trivial rotations, where their trajectory can be continuously contracted to a point, indicating a charge q1256.

Edge/domain-wall states of six-band non-Abelian topological insulators
We now discuss the edge states of the proposed six-band system and the inadequacy of the single-band Zak phase in predicting the existence of edge modes. Figure 1g shows the eigenvalue spectrum and the topological phase diagram obtained by varying a tunable parameter Δ = s = t in the minimal model of Fig. 1a. It displays the eigenvalue spectrum of a 20-unit-cell lattice, where edge and bulk states are distinguished by their inverse participation ratio (IPR). The IPR is defined as IPR = Σ|φi|4, where φi is the normalized wavefunction at lattice i. The states with a large IPR (appear in red) are considered edge states, while those with a small IPR (appearing gray) are considered bulk states. The corresponding IPR threshold is approximately 0.15. The diagram shows five phase regions separated by gap-closing critical points: P1 (q123456), P2 (q16), P3 (q123456), P4 (q1256), and P5 (q123456). The band dispersions shown in Fig. 1b, c correspond respectively to Δ = 0.5 and Δ = 2.5. To provide an intuitive understanding of this phase diagram, Fig. 1h illustrates a 2D parameter plane in momentum space, with each loop representing the Brillouin zone of a one-dimensional Hamiltonian. If the frame rotation of the 1D Hamiltonian along the loop is characterized by a specific generalized quaternion charge, then the loop encloses singularity points in the 2D plane, corresponding to gap-closing points of the Hamiltonian. A phase transition, marked by gap-closing of the Hamiltonian, occurs if the loop crosses these singularity points. We can label these singularity points with the generalized quaternion, indicating that loops that encircle those points are characterized by the same topological charges from a frame rotation perspective. Figure 1i shows representative field patterns of edge states, namely I and II, which are marked in Fig. 1g.

Since non-Abelian topological insulators need to consider multiple band gaps, the edge state distribution in the band structure becomes complicated. We first discuss the case of charges ±qij (i<j). In an ideal model with charge ±qij, where the rotational trivial bands are decoupled from the nontrivial (i-th and j-th) bands (with “nontrivial” referring to eigenvectors that rotate to their antipodal points in parameter space), and therefore the i-th and j-th bands form a two-band subsystem. The gap of this subsystem exhibits a Zak phase of π, while the trivial band has a Zak phase of zero. As a result, the system hosts an odd number of edge states. When perturbations couple the trivial and nontrivial bands, the charge ±qij can be factorized into a series product of charges, and all gaps between the i-th and j-th bands carry a Zak phase of π with an odd number of edge states. For instance, in the topological phase of q16 in Fig. 1g, each gap has one edge state, as q16 can be factorized into q16 = q12 q23 q34 q45 q56. Compared to the quaternion P3 in a three-band system corresponding to 3D rotation, the generalized quaternion P6 in a six-band system exhibits two kinds of unique topological charges: ±qijkl and ±q123456, which have no counterpart in P3. These charges correspond to double and triple rotations involving two and three orthogonal planes, respectively, thus cannot be described using the single band Zak phase description. The Zak phase of π only indicates the rotation angle of π but not the rotation plane, which becomes indeterminate in four or higher-dimensional spaces. Thus, while the Zak phase of π is useful in two-band and three-band systems, it does not provide sufficient information in higher-dimensional systems. Taking the charge q1256 as an example, it can be factorized in special cases into three forms: q12 q56, q16 q25, and -q15 q26. These factorizations can transform continuously without closing gaps, each representing a distinct combination of rotation planes and edge state distributions. In the general case, it is a superposition of these three factorizations and cannot be decomposed into the product of two simple topological charges. The edge state distribution also lies between these three factorization cases. For the charges ±qijkl, there are 15 distinct factorizations (C2C2=3!), leading to more complex edge state distributions. In the phase diagram of Fig. 1g, the edge states lie between various gaps depending on the factorization.

In non-Abelian band topology, the distribution of domain-wall states between two samples is related to the quotient of their topological charges, ΔQ = QL / QR, known as the quotient relation18. Here, QL and QR are the topological charges of the left and right samples. The distribution of domain-wall states is determined by ΔQ. Note that the quotient relation is valid only if the two samples share common gaps, allowing discussion of domain-wall states within the gaps in the original non-Abelian topological charge framework. We apply this principle to six-band non-Abelian topological insulators. In Fig. 2a, two samples with charges q123456 and q1256 are placed together, each with 20 unit cells. The couplings at the interface are averaged from the two samples (Fig. 2b). The Bloch bands for each sample are shown in Fig. 2c, and the energy spectrum in Fig. 2d reveals a single domain-wall state in the gap between the third and fourth bands, consistent with the quotient relation q123456 / q1256 = q34. Representative field patterns of edge and domain-wall states are shown in Fig. 2e. Similarly, for the q123456 and q16 samples (Fig. 2f–j), the energy spectrum (Fig. 2i) shows two domain-wall states in the second-third and fourth-fifth gaps, consistent with q123456 / q16 = q2345.

The quotient relation can be intuitively understood as follows. Similar to Fig. 1h, Fig. 2g displays a parameter plane, where each loop corresponds to a one-dimensional Hamiltonian. As we transition from the left domain q123456 to the right region q16 in real space across the boundary, this process can be represented in parameter space by contracting the loop from a bigger q123456 loop (solid yellow) to a smaller loop q16 (dashed yellow). The q123456 loop encloses charges q2345 and q16, while the q16 loop encloses only q16. As the loop contracts, it passes through q2345, causing the domain-wall state distribution to align with the edge state of this charge. It is important to note that the quotient relation has its own limitations. It is only capable of predicting the types of singularities that the closed loop will pass through, as well as the associated band gap closures, during the continuous transition between two Hamiltonians. Based on the Jackiw-Rebbi-type argument43, domain-wall states appear at gap-closing points, but this applies primarily to points near the gap closing. Therefore, if two samples, together with their domain wall, can be described by a parameter perturbation model around the phase transition point, the quotient relation can accurately predict the distribution of domain-wall states. However, when the parameter differences between the two samples are large, the distribution of domain-wall states becomes highly complex, making the quotient relation unable to directly predict their distribution. In our study, since the parameters of the two samples do not differ significantly, the prediction based on the quotient relation remains effective.

Methods
Numerical simulations of light propagation in the non-Abelian topological insulator
We explain in detail how Fig. 3d, e is obtained from numerical simulations. We take Fig. 3e for example, in which we have considered a total of 205 samples with gout ranging between 0.2 μm and 4.5 μm, while all the samples share the same gin = 1.8 μm. In the simulation of each sample, we model the system by a Hamiltonian H(z) and solve the Schrödinger-like equation H(z)|φ(z)i = -i∂z|φ(z)i, where |φ(z)i is the state vector to be solved. To excite the edge mode II efficiently at the starting point, we choose |φ(z=0)i as the solved eigenvector of the edge mode II, which is given in the right panel of Fig. 3g. Since Fig. 3e only shows the field distributions in the lattice C, we only mark a star at site C2. The coordinates of the star, (1.8 μm, 2), mark the two key excitation parameters: gin = 1.8 μm and the site index n=2 corresponding to excitation site C2 on lattice C. After performing simulations for all the samples, we only extract the output result |φ(z=L)i for each case. By combining |φ(z=L)i of all the 205 simulations, we then draw |φ(z=L)i as functions of gout, which is finally shown in Fig. 3e. This is the procedure for obtaining Fig. 3d, e in numerical simulations. We introduce a detuning Δ to all the gap distances. The phase diagrams of two lattices, namely lattice I with Δ = -0.4 μm and lattice II with Δ = 0.4 μm, are shown on top of Fig. 4b, where the phase transition points are found to shift to a smaller (larger) g with a negative (positive) Δ. Such tunability allows us to combine the two lattices to demonstrate the non-Abelian bulk-boundary correspondence. Figure 4a shows the schematic diagram of a coupled system in which the light grey and dark grey waveguide arrays represent the defined lattices I and II, respectively. By setting the interface gap distance gb = 4.8 μm, we show the calculated eigenvalues of the system as a function of g in Fig. 4b, where the interface modes are marked by red. We calculate the associated non-Abelian topological charge of the combined system by using the non-Abelian quotient relation. The corresponding phase diagram is given on top of Fig. 4b, which is consistent with the distribution of all the interface modes. For example, in the defined phase region P2, the two interface modes are located in the second-third and fourth-fifth gaps, which resulted from the charge q123456 / q16 = q2345.

Sample fabrication and measurement
The photonic waveguide arrays were fabricated using the femtosecond laser direct writing techniques44–47. In our experiment, we focused a laser (Light Conversion CARBIDE 5W) inside a boroaluminosilicate glass chip (with a refractive index of ~1.504) by using a microscope with a numerical aperture of 0.75. To write photonic waveguides following a predesigned trajectory, the movement of the glass chip was controlled by using an Aerotech motion system with a moving speed of 40 mm/s. In the focused region of the laser, a refractive index increment of ~0.0025 can be induced, which then acts as the waveguide core, while the unaffected region acts as the waveguide cladding. The fabricated waveguide exhibits a cross-section of ~6 μm × 6 μm. In such a configuration, the waveguide supports only one vertical-polarized fundamental mode, which is used to probe the physics of the non-Abelian topological insulator. The length L of all the samples is 25 mm.

To effectively excite the topological edge mode and interface mode, we prepared a case-by-case excitation with a light field distribution similar to the eigenfield distribution of the desired mode. For the edge mode I in Fig. 3f, the light energy is mainly localized in waveguides A1 and C1, but the electric fields in the two waveguides are out of phase. To excite this anti-symmetric mode, we first used a Y-branch to split the injected laser into two beams. We then employed a three-waveguide system featuring a stimulated Raman adiabatic passage to generate a geometric phase π for one beam. Meanwhile, we applied another straight waveguide for transporting the other beam without this geometric phase. By connecting the two waveguides carrying these two beams with a phase difference π to waveguides A1 and C1, the edge mode I can be excited effectively. For the edge mode II in Fig. 3g, light is mainly localized in waveguides B2 and C2, while the corresponding electric fields in the two waveguides are in phase. By using a Y-branch beam splitter ahead of the sample, the edge mode II can be excited effectively. The edge mode III was excited using a similar method. For the interface modes I, II, and III, the distributions of light intensity and phase are even more complex, but our excitation method is still effective. Due to the difficulty in ideally exciting these interface modes, our experimental results are not exactly the same as the numerical results, but they still capture the key characteristic of the corresponding interface mode. At the output facet of the sample, we employed a charge-coupled device (XG500, XWJG) to collect the light intensity distributions.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "I have read the paper by Xu-Lin Zhan et al. This concerns a realization of a 6-band non-Abelian insulator in a photonic\nwaveguide array. Given that multi-gap/non-Abelian is a rich and upcoming field I am sympathetic towards publication in this\njournal, also because the 6 band model is realized. However the theory sections are rather sloppy, not the best referenced\nand at some points just plainly inaccurate. I understand this is a well established group in the meta-material community but\nthey should refrain from just putting non-informed theory descriptions. As such, the current form cannot be published.\n1.First of all the homotopy theory [also of the many-band] case was explained in Physical Review B 102 (11), 115135\n(2020). This sets the manner to define the non-Abelian charges. Apart from that this work is not even mentioned, the\ndescription of the quaternion charges in this paper is really sloppy at best. There is a relation with quaternions and rotations,\nsee refs 23 and 25 for instance, but the charges come out as of the classifying space. In 3 bands this is set by O(3)/O(1)^3 which gives a non-trivial first homotopy group. The rotations of the frames then give the manner frame charges transform but\nthis is not the same as just rotations of the 3 band Hamiltonian. Indeed, only in 3D systems the quaternions are directly\nrelated to (third) homotopy invariant, see Ref 25. The authors just say the 6 bands define a double quaternion and then come\nup with how charges relate. This is not defined properly or really related to topological chargers as done for 3 band models.\nThe identification for more bands is known to be the Salingaross group, see above reference as well as the methods and\nappendices of References 9,10. This discussion should be updated.\n2. The same not informed discussion appears in describing the edge states. They simply refer to Ref 14 in lines 176/177. I\nwould urge the authors to read/[reread for the authors that were also involved in Ref 14] the clear issues with just dividing the\ncharge. Note that this in any case only pertains to the Zak phases and edge states as discussed in this system. More\nimportantly I note the discussion on page 41 of the rebuttal file of Refrence 14, see https://static-\ncontent.springer.com/esm/art%3A10.1038/s41586-021-03521-3/MediaObjects/41586_2021_3521_MOESM2_ESM.pdf,\n“Therefore, when comparing two different band structures, one can only tell whether they are in the same or different\nconjugacy class. In contrast, the “absolute” charge difference ΔQ=QL/QR (the quantity purported to govern the “generalized\nbulk-edge correspondence”) seems to be meaningful only if the two models are relatable through a small (gap-closing)\ntuning of parameters, but not in general”. This is illustrated with model and arguments “\nThis should be clearly reflected on. Indeed, this construction only counts Zak phases and also only pertains to models with\nWannier functions at centers. This is is by no means a reliable full proof bulk boundary correspondence and should not be\ncited as such.\n3. The authors state “Additionally, previous studies have mainly focused on static systems with fixed non-Abelian topological\ncharges, while the exploration of dynamic phase transitions between different charges remains unexplored. These\ntransitions hold rich physics that could inspire applications for manipulating topological states.” in lines 57-59 and “In this\nway, dynamic non-Abelian topological phase transitions are directly observed in experiments.”\nI think this is not correct. The parameters are changed over the region; this is in no way dynamical. Several phases as\nfunction of parameters, and change of topological Euler invariant- which is intricate beyond 1D- were shown in Ref 15.\nClaiming a dynamical system when just parameters are changed along the axis is clearly overstating.\nminor comments\nit would important to note that multi-gap and non-Abelian phases arise in magnetic systems, see\nhttps://journals.aps.org/prb/abstract/10.1103/PhysRevB.103.245127 and https://journals.aps.org/prb/abstract/10.1103/xnqg-\n3bgh\nthe title is very general although the real new realisation is a 6-band non-Abelian model\nIn lines 93 and 94 the authors should state that a general manner to make the Hamiltonian real proceeds though the Takagi\ndecomposition", "reviewer_2": "The authors demonstrate the first photonic realization of non-Abelian topological insulators at optical frequencies. They\npropose and carefully analyze a simple model supporting non-Abelian topological charges, consisting essentially of coupled\nSSH chains. They realize the model using laser-written coupled waveguides, and demonstrate both the delocalization of\nedge states across topological phase transitions along the propagation direction, and the existence of interface modes\nbetween adjacent crystals with different topological charge, a consequence of the non-Abelian bulk-boundary\ncorrespondence. The experimental data look clean and are strongly supported by numerical calculations.\nI found the manuscript exceptionally well written, detailed, clear and the content is quite interesting. I therefore recommend\npublication in Nature Communication.\nOnly one minor point:\n- regarding Fig. 3 d-e, I do not understand what the authors mean by “excited as marked by the star”. Only one site is\nexcited? The star refers to the propagation direction and the excitation is provided by the mode eigenstate? Other? The\nexperimental protocol is instead nicely explained in the methods.", "reviewer_3": "At present, non-Abelian topological insulators have been verified in platforms such as metamaterials, transmission line\nnetworks, and acoustic systems. However, in the field of photonics, which has rich application potential, there is still a lack of\nfeasible optical frequency implementation models. Moreover, existing research mostly focuses on static systems with fixed\ntopological charges, and the exploration of dynamic phase transitions between different charges is insufficient. In addition,\nthe traditional Zak phase framework cannot accurately predict the distribution of edge states in multi-band systems. It is\nnecessary to solve the correlation problem between topological invariants and edge states in multi-band PT-symmetric\nsystems based on non-Abelian band topology theory.\nIn response to the above issues, this study proposes a minimum model of non-Abelian topological insulators. A six-band\nnon-Abelian topological insulator is experimentally achieved through a photonic waveguide array at optical frequencies to\nexplore dynamic non-Abelian topological phase transitions and verify the domain wall state distribution dominated by non-\nAbelian quotient relations. At the same time, the model is extended to a four-band system to prove its universality. However,\nI think the current version is not suitable for acceptance by Nature Communications. Some modifications are necessary\nbefore acceptance.\nMy comments are:\n1.Although the supplementary materials provide the multiplication table of Q_{64}, they do not elaborate on the numerical\ncalculation process of the generalized quaternion charge Q, such as the selection criteria for the number of discrete k points\nand the numerical implementation method of the path ordered product. And how to map elements from $\\mathfrak{spin}(N)$\nLie algebras to elements of generalized quaternion groups makes it difficult for readers to reproduce the calculation of\ntopological charges. It is suggested that the author take the four-band model as an example to illustrate how to obtain the\ncorresponding results.\n2.The topological protection origin of non-Abelian topological charges was not explicitly expounded, such as its association\nwith system symmetries (PT symmetry, time-reversal symmetry).\n3.What is the clear definition of the topological charge ?\n4.What is the relationship between the elements of group Q64 and group Q16 and the topological edge states? How to\nanalyze this connection?\n5.Although the supplementary materials present the research results of the four-band system, they do not systematically\ncompare the commonalities and differences between the four-band and six-band systems in terms of topological charge\nclassification, phase transition rules, and edge state characteristics, such as whether an increase in the number of bands will\nintroduce new topological phases, which fails to reflect the regularity of model expansion.\n6.The article mentions the distinction between bulk states and edge states through the inverse participation ratio (IPR), but\ndoes not provide a specific definition of IPR (nor does it explain the selection criteria for the IPR threshold), resulting in an\nunclear basis for distinguishing marginal states from postures).\n7.The experiment observes the phase transition by changing the gap $g_{out}$ at the output end of the sample. Essentially,\nit is a static comparative experiment rather than a real-time dynamic observation, which cannot capture the evolution\ndynamics of the topological state during the phase transition process (such as the time scale of edge mode disappearance),\nthus limiting the in-depth understanding of the non-Abelian phase transition mechanism. This also goes against the authors'\noriginal intention, as in the introduction, the authors explicitly mentioned the study of dynamical phase transitions.\n8.What is the motivation for choosing s=t to conduct research? What changes will occur in the result if s is different from t?"}, "author_response": "Responses to Reviewer 1\nI have read the paper by Xu-Lin Zhan et al. This concerns a realization of a 6-band non-Abelian insulator in\na photonic waveguide array. Given that multi-gap/non-Abelian is a rich and upcoming field I am sympathetic\ntowards publication in this journal, also because the 6 band model is realized. However, the theory sections\nare rather sloppy, not the best referenced and at some points just plainly inaccurate. I understand this is a\nwell established group in the meta-material community but they should refrain from just putting non-\ninformed theory descriptions. As such, the current form cannot be published.\nWe thank the reviewer for his/her efforts in reviewing our work. We also appreciate the positive opinion\nfrom the reviewer as he/she is “sympathetic towards publication”. In what follows, we will follow the\nreviewer’s suggestions to revise the theoretical part of our manuscript.\nComment 1.1:\n(1) First of all the homotopy theory [also of the many-band] case was explained in Physical Review B 102\n(11), 115135 (2020). This sets the manner to define the non-Abelian charges. Apart from that this work is\nnot even mentioned, the description of the quaternion charges in this paper is really sloppy at best. There is\na relation with quaternions and rotations, see refs 23 and 25 for instance, but the charges come out as of the\nclassifying space. In 3 bands this is set by O(3)/O(1)^3 which gives a non-trivial first homotopy group. The\nrotations of the frames then give the manner frame charges transform but this is not the same as just rotations\nof the 3 band Hamiltonian. Indeed, only in 3D systems the quaternions are directly related to (third)\nhomotopy invariant, see Ref 25. The authors just say the 6 bands define a double quaternion and then come\nup with how charges relate. This is not defined properly or really related to topological chargers as done for\n3 band models. The identification for more bands is known to be the Salingaros group, see above reference\nas well as the methods and appendices of References 9,10. This discussion should be updated.\nResponse:\nWe thank the reviewer for the suggestions. We apologize for the omission of the description of the classifying\nspace. For our six-band PT-symmetric system, all bands are non-degenerate, and the corresponding\nHamiltonian order-parameter space is 𝑀 =𝑂(6)/𝑂(1)6 = SO(6)/ℤ5. Here, 𝑂(6) is the 6D orthogonal\n6 2\ngroup and 𝑂(1)6 ≅ℤ6 arises from the ±1 gauge freedom of the eigenvectors. 𝑆𝑂(6) represents the 6D\n2\nspecial orthogonal group, while ℤ5 corresponds to the 𝜋 rotation in 5 independent 2D planes: [𝜓 ,𝜓 ],\n2 1 2\n[𝜓 ,𝜓 ], [𝜓 ,𝜓 ], [𝜓 ,𝜓 ] and [𝜓 ,𝜓 ] , where 𝜓 is the 𝑗-th eigenstate. Therefore, the fundamental\n1 3 1 4 1 5 1 6 𝑗\nhomotopy group of the corresponding Hamiltonian space is the Salingaros vee group 𝜋 (𝑀 )=𝑃̅ =\n1 6 6\n⋃ {±𝑒 𝑛1𝑒 𝑛2⋯𝑒 𝑛5},\nformed by products of the basis elements of the real Clifford algebra 𝐶ℓ :\n𝑛𝑖∈{0,1} 1 2 5 0,5\n{𝑒 ,𝑒 ,𝑒 ,𝑒 ,𝑒 }. Here, 𝑒 corresponds to the 𝜋 rotation in the plane of [𝜓 ,𝜓 ].\n1 2 3 4 5 𝑗 1 𝑗+1\nIn the previous manuscript, we imprecisely termed the Salingaros vee group as a generalized quaternion\ngroup, following the terminology used in [Science 365, 1273 (2019)], without providing an accurate\nexplanation and reference. Considering that the quaternion group admits many different generalizations and\nthat 𝑄 has been used in other literature to denote alternative types of generalized quaternion group, we\n64\nhave revised our manuscript to refer to the homotopy group more precisely as the Salingaros vee group and\nhave adopted the 𝑃̅ notation (following the Science paper). Moreover, in the revised manuscript, we have\n𝑁\nincluded the paper “Physical Review B 102 (11), 115135 (2020)” as Ref. 12.\nAbelian topological insulator model with six bands” in the main text and changed the corresponding\ndescription in the Supplementary Note 3 of the Supplementary Information.\nParagraph 3 of the section “Minimal non-Abelian topological insulator model with six bands”:\n“Considering all the bandgaps between adjacent bands, the order-parameter space of the Hamiltonian is\n𝑀 =𝑂(6)/𝑂(1)6, where 𝑂(6) is the 6D orthogonal group and 𝑂(1)6 ≅ℤ6 arises from the ±1 gauge\n6 2\nfreedom of the eigenvectors. The corresponding fundamental homotopy group is given by the Salingaros vee\ngroup1 that 𝜋 (𝑀 )=𝑃̅ =⋃ {±𝑒 𝑛1𝑒 𝑛2⋯𝑒 𝑛5}, which is formed by products of the basis elements\n1 6 6 𝑛𝑖∈{0,1} 1 2 5\nof the real Clifford algebra 𝐶ℓ : {𝑒 ,𝑒 ,𝑒 ,𝑒 ,𝑒 }, where {𝑒 ,𝑒 }=−2𝛿 .”\n0,5 1 2 3 4 5 𝑖 𝑗 𝑖𝑗\nParagraph 3 of the Supplementary Note 3 “Minimal four-band non-Abelian topological insulator\nmodel”: “Considering all the gaps between neighboring bands, the order-parameter space of the gapped\nHamiltonian is 𝑀 =𝑂(4)/𝑂(1)4, where 𝑂(4) is the 4D orthogonal group and 𝑂(1)4 ≅ℤ4 arises from\n4 2\nthe ±1 gauge freedom of the eigenvectors. The corresponding fundamental homotopy group is given by the\nSalingaros vee group that 𝜋 (𝑀 )=𝑃̅ =⋃ {±𝑒 𝑛1𝑒 𝑛2𝑒 𝑛3}, where 𝑒 , 𝑒 , 𝑒 are the basis of real\n1 4 4 𝑛𝑖∈{0,1} 1 2 3 1 2 3\nClifford algebra 𝐶ℓ , satisfying {𝑒 ,𝑒 }=−2𝛿 .”\n0,3 𝑖 𝑗 𝑖𝑗\nComment 1.2:\n(2) The same not informed discussion appears in describing the edge states. They simply refer to Ref 14 in\nlines 176/177. I would urge the authors to read/[reread for the authors that were also involved in Ref 14] the\nclear issues with just dividing the charge. Note that this in any case only pertains to the Zak phases and edge\nstates as discussed in this system. More importantly I note the discussion on page 41 of the rebuttal file of\nRefrence 14, see https://static-content.springer.com/esm/art%3A10.1038/s41586-021-03521-\n3/MediaObjects/41586_2021_3521_MOESM2_ESM.pdf.\n“Therefore, when comparing two different band structures, one can only tell whether they are in the same or\ndifferent conjugacy class. In contrast, the “absolute” charge difference ΔQ=QL/QR (the quantity purported\nto govern the “generalized bulk-edge correspondence”) seems to be meaningful only if the two models are\nrelatable through a small (gap-closing) tuning of parameters, but not in general”. This is illustrated with\nmodel and arguments”.\nThis should be clearly reflected on. Indeed, this construction only counts Zak phases and also only pertains\nto models with Wannier functions at centers. This is by no means a reliable full proof bulk boundary\ncorrespondence and should not be cited as such.\nResponse:\nWe thank the reviewer for this question. We agree with the reviewer that the quotient relation ∆𝑄 =𝑄 /𝑄\n𝐿 𝑅\nis not a generally established bulk-edge correspondence. It is typically effective in determining the\ndistribution of domain-wall states when the parameters of the two samples are near the same gap-closing\npoint. However, for two samples with significantly different parameters, the distribution of domain-wall\nstates becomes much more complex and cannot be directly determined using the quotient relation. The value\nof this quotient relation, ∆𝑄 =𝑄 /𝑄 , lies in identifying the types of singularities that the corresponding\n𝐿 𝑅\nclosed loop will encounter during the continuous parameter transition from one Hamiltonian to another. This,\nin turn, corresponds to the types of band gap closures that the band structure will undergo. According to the\nJackiw-Rebbi type argument1, domain-wall states will appear around the gap-closing points. If the\nparameters continue to change, even without additional gap closing, the domain-wall states may gradually\n2\nshift to other gaps, or merge into the energy region of the bulk states, coupling with the bulk states and\neventually disappearing. This is why ∆𝑄 =𝑄 /𝑄 cannot serve as a universal predictor for the distribution\n𝐿 𝑅\nof domain-wall states. In our study, since the parameters of the two samples do not differ significantly, the\nprediction based on the quotient relation remains effective.\nIn the revised manuscript, we have included the clarification in the section of “Edge/domain-wall\nstates of six-band non-Abelian topological insulators” in the main text.\nParagraph 5 of the section “Edge/domain-wall states of six-band non-Abelian topological\ninsulators”: “It is important to note that the quotient relation has its own limitations. It is only capable of\npredicting the types of singularities that the closed loop will pass through, as well as the associated band\ngap closures, during the continuous transition between two Hamiltonians. Based on the Jackiw-Rebbi-type\nargument1, domain-wall states appear at gap-closing points, but this applies primarily to points near the\ngap closing. Therefore, if two samples, together with their domain wall, can be described by a parameter\nperturbation model around the phase transition point, the quotient relation can accurately predict the\ndistribution of domain-wall states. However, when the parameter differences between the two samples are\nlarge, the distribution of domain-wall states becomes highly complex, making the quotient relation unable\nto directly predict their distribution. In our study, since the parameters of the two samples do not differ\nsignificantly, the prediction based on the quotient relation remains effective.”\nComment 1.3:\n(3) The authors state “Additionally, previous studies have mainly focused on static systems with fixed non-\nAbelian topological charges, while the exploration of dynamic phase transitions between different charges\nremains unexplored. These transitions hold rich physics that could inspire applications for manipulating\ntopological states.” in lines 57-59 and “In this way, dynamic non-Abelian topological phase transitions are\ndirectly observed in experiments.” I think this is not correct. The parameters are changed over the region;\nthis is in no way dynamical. Several phases as function of parameters, and change of topological Euler\ninvariant- which is intricate beyond 1D- were shown in Ref 15. Claiming a dynamical system when just\nparameters are changed along the axis is clearly overstating.\nResponse:\nWe thank the reviewer for this comment. The reason we claim our system as a “dynamical” system is that\nlight propagating in the paraxial waveguides is governed by 𝐻(𝑧)|𝜑(𝑧)⟩=−𝑖𝜕 |𝜑(𝑧)⟩, where 𝐻(𝑧) and\n𝑧\n|𝜑(𝑧)⟩ are the Hamiltonian and state vector, respectively. The form of the master equation is in fact a\ntime-dependent Schrödinger-like equation in which the original time term t is replaced by a space dimension\nterm z. In our experiment, as the gap distance (which determines the topological charge) varies along the z\naxis, light propagating in the system would experience multiple non-Abelian topological phases at different\n“timeslots” 𝑧. Since different topological phases support different distributions of edge states, measuring the\nexistence of edge states in such “dynamical” systems would help us identify different topological phases and\nthe phase transition between them. This is motivation that we design such a system with varying structure\nparameters.\nWe agree with the reviewer that referring to this system with z-dependent structure parameters as a\n“dynamical” system may not be the most appropriate terminology. A real dynamical system should be one\nthat the Hamiltonian can be tuned in situ such that the associated topological phase can be reconfigured.\nin the Abstract, main text and Supplementary Information, and rewritten the corresponding contents. Some\nkey revisions are listed below for the reviewer to check.\nAbstract: “Different non-Abelian topological charges have been experimentally demonstrated by probing\nthe adiabatic evolution of edge states in a system with varying structural parameters.”\nParagraph 2 in Introduction: “We design and implement an experimental configuration in which light\ncould encounter structures with different non-Abelian topological phases along the propagation direction.\nIn this way, different non-Abelian topological charges can be revealed in experiments.”\nParagraph 3 of the section “Experimental demonstration of edge states associated with different non-\nAbelian topological charge”: “In experiments, the phase transition cannot be directly observed, as it occurs\nduring the light evolution process. Instead, this setting can be employed to distinguish different topological\nphases, as crossing the phase transition point could affect the feature of the excited edge states.”\nComment 1.4:\n(4) It would important to note that multi-gap and non-Abelian phases arise in magnetic systems, see\nhttps://journals.aps.org/prb/abstract/10.1103/PhysRevB.103.245127 and\nhttps://journals.aps.org/prb/abstract/10.1103/xnqg-3bgh.\nThe title is very general although the real new realisation is a 6-band non-Abelian model.\nIn lines 93 and 94 the authors should state that a general manner to make the Hamiltonian real proceeds\nthough the Takagi decomposition.\nResponse:\nWe appreciate these minor comments by the reviewer. We have made the following revision.\n(1) The two references are closely related to this work. So they have been cited as Refs. 25 and 34 in the\nrevised manuscript.\n(2) We agree with the reviewer that the title is too general. We have changed the title to a more specific one:\nPhotonic non-Abelian topological insulators of six bands.\n(3) We have added a sentence to mention the Takagi decomposition in the revied manuscript.\nParagraph 3 of the section “Minimal non-Abelian topological insulator model with six bands”: “The\nunitary matrix 𝑉 can be derived from Takagi’s decomposition 𝑈 =𝑉𝑉𝑇, where 𝑈 is a symmetric matrix\nrepresenting the parity inversion operation.”\nThe authors demonstrate the first photonic realization of non-Abelian topological insulators at optical\nfrequencies. They propose and carefully analyze a simple model supporting non-Abelian topological charges,\nconsisting essentially of coupled SSH chains. They realize the model using laser-written coupled waveguides,\nand demonstrate both the delocalization of edge states across topological phase transitions along the\npropagation direction, and the existence of interface modes between adjacent crystals with different\ntopological charge, a consequence of the non-Abelian bulk-boundary correspondence. The experimental data\nlook clean and are strongly supported by numerical calculations.\nI found the manuscript exceptionally well written, detailed, clear and the content is quite interesting. I\ntherefore recommend publication in Nature Communication.\nWe thank the reviewer for his/her efforts in reviewing our work and the recommendation for publication.\nThe remarks will be addressed below.\nComment 2.1:\nOnly one minor point: Regarding Fig. 3 d-e, I do not understand what the authors mean by “excited as marked by the star”. Only one site is\nexcited? The star refers to the propagation direction and the excitation is provided by the mode eigenstate? Other? The\nexperimental protocol is instead nicely explained in the methods.\nResponse:\nWe feel sorry for confusing the reviewer. In what follows, we will explain in detail how Fig. 3d,e is obtained\nfrom numerical simulations. Our system consists of a waveguide array (Fig. 3a and 3b) in which the vertical\ngap between adjacent row of waveguides varies along the light-propagation direction (the z-direction). The\ngap distance along z-direction is denoted by 𝑔(𝑧) and as 𝑔(𝑧) is varying, 𝑔 ≠𝑔 . We take Fig. 3e for\nout in\nexample and replot the figure in the following Fig. R1d. In the simulation, we consider a total of 205 samples\nwith 𝑔 ranging between 0.2 μm to 4.5 μm, while all the samples share the same 𝑔 = 1.8 μm. In the\nout in\nsimulation of each sample, we model the system by a Hamiltonian 𝐻(𝑧) and solve the Schrödinger-like\nequation 𝐻(𝑧)|𝜑(𝑧)⟩=−𝑖𝜕 |𝜑(𝑧)⟩, where |𝜑(𝑧)⟩ is the state vector to be solved. To excite the edge mode\n𝑧\nII efficiently at the starting point, we choose |𝜑(𝑧=0)⟩ according to the eigenvector of the edge mode II,\nwhich is plotted in Fig. R1e. We find that the edge mode II exhibits a complex field distribution. Therefore,\nthe excitation is not via only one site, but for this case, sites A , B and C are simultaneously excited with\n3 2 2\ncorresponding phases. In other words, the excitation is provided by the mode eigenstate, as expected by the\nreviewer. Since Fig. 3e (or Fig. R1d) only shows the field distributions in the lattice C, we only mark a star\nat site C . The coordinates of the star, i.e., (1.8 μm,2), mark the two key excitation parameters: 𝑔 =\n2 𝑖𝑛\n1.8 μm and the site index 𝑛 =2 corresponding to excitation site C on lattice C. After performing\n2\nsimulations for the 205 samples, we extract the output result, i.e., |𝜑(𝑧 =𝐿)⟩ for each case. Figure R1a-c\nshows the calculated results of |𝜑(𝑧)⟩ in lattice C for three cases with 𝑔 = 1 μm, 2 μm and 3 μm,\nout\nrespectively. The outputs |𝜑(𝑧=𝐿)⟩ are marked by the red, blue and orange dashed box, respectively. By\ncombining |𝜑(𝑧=𝐿)⟩ of all the 205 simulations, we then draw |𝜑(𝑧 =𝐿)⟩ as functions of 𝑔 , which is\nout\nshown in Fig. R1d. This is the procedure we obtain Fig. 3d,e in numerical simulations.\nIn the revised manuscript, we have added the above details in Methods: “Numerical simulations of\nlight propagation in the non-Abelian topological insulator. We explain in detail how Fig. 3d,e is obtained\n5\nfrom numerical simulations. We take Fig. 3e for example, in which we have considered a total of 205 samples\nwith 𝑔 ranging between 0.2 μm to 4.5 μm, while all the samples share the same 𝑔 = 1.8 μm. In the\n𝑜𝑢𝑡 𝑖𝑛\nsimulation of each sample, we model the system by a Hamiltonian 𝐻(𝑧) and solve the Schrödinger-like\nequation 𝐻(𝑧)|𝜑(𝑧)⟩=−𝑖𝜕 |𝜑(𝑧)⟩, where |𝜑(𝑧)⟩ is the state vector to be solved. To excite the edge mode\n𝑧\nII efficiently at the starting point, we choose |𝜑(𝑧 =0)⟩ as the solved eigenvector of the edge mode II,\nwhich is given in the right panel of Fig. 3g. Since Fig. 3e only shows the field distributions in the lattice C, we only\nmark a star at site C . The coordinates of the star, (1.8 𝜇𝑚,2), mark the two key excitation parameters:\n2\n𝑔 =1.8 𝜇𝑚 and the site index 𝑛 =2 corresponding to excitation site C on lattice C. After performing\n𝑖𝑛 2\nsimulations for all the samples, we only extract the output result, i.e., |𝜑(𝑧 =𝐿)⟩ for each case. By\ncombining |𝜑(𝑧 =𝐿)⟩ of all the 205 simulations, we then draw |𝜑(𝑧=𝐿)⟩ as functions of 𝑔 , which is\n𝑜𝑢𝑡\nfinally shown in Fig. 3e. This is the procedure for obtaining Fig. 3d,e in numerical simulations.”\nFig. R1| Details of numerical simulations. a-c The calculated light propagation dynamics in a sample with\n𝑔 = 1 μm (a), 2 μm (b) and 3 μm (c), while 𝑔 = 1.8 μm is fixed. d The simulated light intensity\nout in\ndistribution at the output port of the waveguide arrays as a function of 𝑔 , where the edge mode II is excited\nout\nat the input port with 𝑔 = 1.8 μm (marked by the star). e Eigenfield distribution of the edge mode II which\nin\nis used as the excitation in (a-d).\nAt present, non-Abelian topological insulators have been verified in platforms such as metamaterials,\ntransmission line networks, and acoustic systems. However, in the field of photonics, which has rich\napplication potential, there is still a lack of feasible optical frequency implementation models. Moreover,\nexisting research mostly focuses on static systems with fixed topological charges, and the exploration of\ndynamic phase transitions between different charges is insufficient. In addition, the traditional Zak phase\nframework cannot accurately predict the distribution of edge states in multi-band systems. It is necessary to\nsolve the correlation problem between topological invariants and edge states in multi-band PT-symmetric\nsystems based on non-Abelian band topology theory.\nIn response to the above issues, this study proposes a minimum model of non-Abelian topological insulators.\nA six-band non-Abelian topological insulator is experimentally achieved through a photonic waveguide\narray at optical frequencies to explore dynamic non-Abelian topological phase transitions and verify the\ndomain wall state distribution dominated by non-Abelian quotient relations. At the same time, the model is\nextended to a four-band system to prove its universality. However, I think the current version is not suitable\nfor acceptance by Nature Communications. Some modifications are necessary before acceptance.\nWe thank the reviewer for his/her efforts in reviewing our work and the positive remarks. The detailed\ncomments will be addressed below.\nComment 3.1:\n1. Although the supplementary materials provide the multiplication table of Q_{64}, they do not elaborate\non the numerical calculation process of the generalized quaternion charge Q, such as the selection criteria\nfor the number of discrete k points and the numerical implementation method of the path ordered product. And how to map elements from $\\mathfrak{spin}(N)$ Lie algebras to elements of generalized quaternion\ngroups makes it difficult for readers to reproduce the calculation of topological charges. It is suggested that\nthe author take the four-band model as an example to illustrate how to obtain the corresponding results.\nResponse:\nWe thank the reviewer for this comment. The key step in the non-Abelian charge calculation is to lift the\nBerry-Wilczek-Zee (BWZ) connection one-form from 𝔰𝔬(𝑁) to 𝔰𝔭𝔦𝔫(𝑁) basis2-4. Here we will take the four-\nband Hamiltonian as an example to illustrate the calculation details. The basis of Lie algebra 𝔰𝔬(4) is set as\n(𝐿 ) =−𝛿 𝛿 +𝛿 𝛿 , the explicit forms are\n𝑖𝑗 𝑖𝑎 𝑗𝑏 𝑖𝑏 𝑗𝑎\n𝑎𝑏\n0 −1 0 0 0 0 −1 0 0 0 0 −1\n1 0 0 0 0 0 0 0 0 0 0 0\n𝐿 =( ), 𝐿 =( ), 𝐿 =( ),\n12 0 0 0 0 13 1 0 0 0 14 0 0 0 0\n0 0 0 0 0 0 0 0 1 0 0 0\n0 0 0 0 0 0 0 0 0 0 0 0\n0 0 −1 0 0 0 0 −1 0 0 0 0\n𝐿 =( ), 𝐿 =( ), 𝐿 =( ). (R1)\n23 0 1 0 0 24 0 0 0 0 34 0 0 0 −1\n0 0 0 0 0 1 0 0 0 0 1 0\n𝑖𝑗 4 𝑖 𝑗 𝑖 𝑖=1\nmatrices:\n0 0 0 𝑖 0 0 0 1\n0 0 𝑖 0 0 0 −1 0\nΓ =( ), Γ =( ),\n1 0 −𝑖 0 0 2 0 −1 0 0\n−𝑖 0 0 0 1 0 0 0\n0 0 𝑖 0 0 0 1 0\n0 0 0 −𝑖 0 0 0 1\nΓ =( ), Γ =( ). (R2)\n3 −𝑖 0 0 0 4 1 0 0 0\n0 𝑖 0 0 0 1 0 0\nIn the numerical calculation, the BWZ connection is obtained through [𝐴 (𝒌)]𝑖 =⟨𝑢𝑖|𝜕 |𝑢 𝑗 ⟩ and can\n𝑎𝑙𝑙 𝛼𝑗 𝒌 𝑘𝛼 𝒌\nbe decomposed into the 𝔰𝔬(4) basis as [𝐴 (𝒌)] =∑ 𝛽 𝑖𝑗 𝐿 . After obtaining the coefficients 𝛽 𝑖𝑗 , the\n𝑎𝑙𝑙 𝛼 𝑖<𝑗 𝛼 𝑖𝑗 𝛼\nBWZ connection can be lifted onto the 𝔰𝔭𝔦𝔫(𝑁)-valued 1-form as [𝐴̅ (𝒌)] =∑ 𝛽 𝑖𝑗 𝑡 . Then, the\n𝑎𝑙𝑙 𝛼 𝑖<𝑗 𝛼 𝑖𝑗\nmatrix-formed non-Abelian charge is calculated using the path-ordered product, given by\n𝑄 =∏ exp (𝐴̅ (𝒌 )∙Δ𝒌). (R3)\n{𝒌𝛼}\n𝛼\nΛ\n=1\n𝑎𝑙𝑙 𝛼\nNext, we need to compare the path-ordered product 𝑄 with the elements of the 𝑄 group to determine which\n16\ncharge it corresponds to. To comply with the first reviewer's suggestions, we have renamed 𝑸 and\n𝟏𝟔\n𝑸 as 𝑷̅ and 𝑷̅ respectively throughout the manuscript (see reply to Comment 1.1). The elements of\n𝟔𝟒 𝟒 𝟔\nthe 𝑃̅ group are generated by the basis of Clifford algebra 𝐶ℓ , which is\n4 0,3\n1\n𝑒 = [Γ,Γ ] (R4)\n𝑗−1 𝑗 1\n2\nfor 2≤𝑗 ≤𝑁. The matrix forms are\n−𝑖 0 0 0\n1 0 𝑖 0 0\n𝑒 = [Γ ,Γ ]=( ),\n1 2 2 1 0 0 −𝑖 0\n0 0 0 𝑖\n0 1 0 0\n1 −1 0 0 0\n𝑒 = [Γ ,Γ ]=( ),\n2 2 3 1 0 0 0 1\n0 0 −1 0\n0 −𝑖 0 0\n1 −𝑖 0 0 0\n𝑒 = [Γ ,Γ ]=( ). (R5)\n3 2 4 1 0 0 0 𝑖\n0 0 𝑖 0\nThen the 𝑃̅ group can be expressed as\n4\n𝑃̅ = ⋃ {±𝑒 𝑛1𝑒 𝑛2𝑒 𝑛3}\n4 1 2 3\n𝑛𝑖∈{0,1}\n= {±𝑒 ,±𝑒 ,±𝑒 ,±𝑒 𝑒 ,±𝑒 𝑒 ,±𝑒 𝑒 }∪{+1,−1,+𝑒 𝑒 𝑒 ,−𝑒 𝑒 𝑒 }, (R6)\n1 2 3 1 2 1 3 2 3 1 2 3 1 2 3\nwhere\n0 −𝑖 0 0 0 −1 0 0\n−𝑖 0 0 0 1 0 0 0\n𝑒 𝑒 =( ), 𝑒 𝑒 =( ),\n1 2 0 0 0 −𝑖 1 3 0 0 0 1\n0 0 −𝑖 0 0 0 −1 0\n8\n−𝑖 0 0 0 −1 0 0 0\n0 𝑖 0 0 0 −1 0 0\n𝑒 𝑒 =( ), 𝑒 𝑒 𝑒 =( ). (R7)\n2 3 0 0 𝑖 0 1 2 3 0 0 1 0\n0 0 0 −𝑖 0 0 0 1\nAfter comparing the path-ordered product 𝑄 with the elements of the 𝑃̅ group one by one, if the element\n4\nsummation of their matrix difference is smaller than a threshold, i.e., ∑ |𝑄−𝑞 | <𝛿, then 𝑄 is\n𝑖,𝑗 𝑙 𝑖𝑗\nconsidered equal to 𝑞 , where 𝑞 is the 𝑙-th elements in the 𝑃̅ group. The criteria for determining the number\n𝑙 𝑙 4\nof discretized 𝒌-points is whether there exists an element in 𝑃̅ such that ∑ |𝑄−𝑞 | <𝛿. If it does not\n4 𝑖,𝑗 𝑙 𝑖𝑗\nexist, we will increase the number of 𝒌-points. For the four-band calculation in the Supplementary\nInformation, we set the number Λ of 𝒌-points as 2,000 and the threshold as 0.2. For the six-band calculation\nin the main text, we set the number Λ of 𝒌-points as 10,000 and the threshold as 0.2.\nIn the revised Supplementary Information, we have added more details about the charge calculation in\nSupplementary Note 2.\nParagraph 1 of the Supplementary Note 2 “Calculation method and multiplication table of 𝑷̅ ”: “After\n𝟔\ncomparing the path-ordered product 𝑄 with each element in the generalized quaternion group 𝑃̅ one by\n𝑁\none, if the sum of their matrix difference is less than a threshold, i.e., ∑ |𝑄−𝑞 | <𝛿, then 𝑄 is\n𝑖,𝑗 𝑙 𝑖𝑗\nconsidered equal to 𝑞 . Here, 𝑞 is the 𝑙-th elements in 𝑃̅ . The discretized 𝒌-points sampling is considered\n𝑙 𝑙 𝑁\nsufficient if an element in 𝑃̅ exists and satisfies ∑ |𝑄−𝑞 | <𝛿.”\n𝑁 𝑖,𝑗 𝑙 𝑖𝑗\nComment 3.2:\n2. The topological protection origin of non-Abelian topological charges was not explicitly expounded, such\nas its association with system symmetries (PT symmetry, time-reversal symmetry).\nResponse:\nWe thank the reviewer for this question. Non-abelian topological charges were usually used to study systems\nwith PT symmetry, characterizing the topological properties of eigenvectors in momentum space. PT\nsymmetry ensures that the system remains invariant under the combined operations of spatial inversion (P)\nand time reversal (T). In spinless systems, after choosing an appropriate basis, the PT operator can be\nrepresented by the complex conjugation operator 𝐾. Therefore, in this basis, the Hamiltonian always\nsatisfies 𝐻(𝐤) = 𝐻∗(𝐤), making it a real matrix for any Bloch wavevector 𝐤. The appropriate basis can be\nobtained through unitary transformation 𝐻(𝐤)=𝑉∙𝐻 (𝐤)∙𝑉†. The unitary matrix 𝑉 can be derived from\noperation. The real Hamiltonian ensures real eigenvectors across the entire Brillouin zone, thereby\ndetermining the Hamiltonian space for topological classification.\nThe topological properties of a two-band PT-symmetric system can be described by Abelian topological\ncharges (such as integer or ℤ classification). In the case of multiple bands, if only one band gap is\n2\nconsidered, the topological classification still falls within the Abelian category, for example 𝜋 (𝑀 )=\n1 𝑚,𝑛\nℤ , where 𝑀 = 𝑂(𝑚+𝑛)/𝑂(𝑚)×𝑂(𝑛), with 𝑂(𝑚) representing the orthogonal group, and 𝑚 and 𝑛\n2 𝑚,𝑛\nreferring to the number of conduction and valence bands, respectively. When multiple band gaps are\ninvolved, the topological classification of the system requires the use of non-Abelian topological charges.\nFor a PT-symmetric non-degenerate 𝑁-band system, the order parameter space of the Hamiltonian can be\nrepresented as 𝑀 =𝑂(𝑁)/𝑂(1)𝑁. The fundamental homotopy group of this structure corresponds to the\n𝑁\nthe Salingaros vee group2 that 𝜋 (𝑀 )=𝑃̅ =⋃ {±𝑒 𝑛1𝑒 𝑛2⋯𝑒 𝑛𝑁−1}. For a 3-band system, the\n1 𝑁 𝑁 𝑛𝑖∈{0,1} 1 2 𝑁−1\ncorresponding group is a quaternion group 𝜋 (𝑀 )=𝑃̅ ={+1,±𝑖,±𝑗,±𝑘,−1}, which is a non-Abelian\n1 3 3\n9\ngroup consisting of five conjugacy classes and eight group elements, satisfying the relations 𝑖𝑗 =𝑘, 𝑗𝑘 =\n𝑖,𝑘𝑖 =𝑗, and 𝑖2 =𝑗2 =𝑘2 =−1. The group element relations clearly show that for 𝑖, 𝑗, 𝑘, the order of\nmultiplication influences the result. For instance, 𝑖𝑗 =𝑘, but 𝑗𝑖 =−𝑘, illustrating the non-commutative\nproperty of the non-Abelian group.\nIn the revised manuscript, we have added more descriptions to explain the topological protection origin\nof the non-Abelian topological charges in the main text.\nParagraph 1 of the Introduction: “Here, PT symmetry constrains the Hamiltonian to remain real across\nthe Brillouin zone, which facilitates topological characterization within a Clifford algebra framework.”\nParagraph 3 of the section “Minimal non-Abelian topological insulator model with six bands”: “The\nunitary matrix 𝑉 can be derived from Takagi’s decomposition 𝑈 =𝑉𝑉𝑇, where 𝑈 is a symmetric matrix\nrepresenting the parity inversion operation.”\n“Considering all the bandgaps between adjacent bands, the order-parameter space of the Hamiltonian is\n𝑀 =𝑂(6)/𝑂(1)6, where 𝑂(6) is the 6D orthogonal group and 𝑂(1)6 ≅ℤ6 arises from the ±1 gauge\n6 2\nfreedom of the eigenvectors. The corresponding fundamental homotopy group is given by the Salingaros vee\ngroup1 that 𝜋 (𝑀 )=𝑃̅ =⋃ {±𝑒 𝑛1𝑒 𝑛2⋯𝑒 𝑛5}, which is formed by products of the basis elements\n1 6 6 𝑛𝑖∈{0,1} 1 2 5\nof the real Clifford algebra 𝐶ℓ : {𝑒 ,𝑒 ,𝑒 ,𝑒 ,𝑒 }, where {𝑒 ,𝑒 }=−2𝛿 .”\n0,5 1 2 3 4 5 𝑖 𝑗 𝑖𝑗\n“The topological phase classification is determined by this non-Abelian fundamental group 𝑃̅ , and the\n6\ngroup elements are referred to as the non-Abelian topological charges characterizing the topological\nphases.”\nComment 3.3:\n3. What is a clear definition of the topological charge?\nResponse:\nWe thank the reviewer for this question. As discussed in our reply to Comments 1.1 and 3.2, the order\nparameter space of the 𝑁-band non-degenerate Hamiltonian is 𝑀 =𝑂(𝑁)/𝑂(1)𝑁, and the topological\n𝑁\nphase classification is determined by the corresponding fundamental homotopy group, 𝜋 (𝑀 )=𝑃̅ . This\n1 𝑁 𝑁\nfundamental homotopy group can be identified by the real Clifford algebra 𝐶ℓ , which is referred to\n0,𝑁−1\nas the Salingaros vee group 𝑃̅ =⋃ {±𝑒 𝑛1𝑒 𝑛2…𝑒 𝑛𝑁−1}. It can be understood from the perspective of\n𝑁 𝑛𝑖∈{0,1} 1 2 𝑁−1\nliquid crystal defects or band degeneracies, offering insight into their topological properties. 𝑃̅ is a non-\n𝑁\nAbelian group for 𝑁 ≥3, and the non-Abelian topological charges are given by the group elements of\n𝑷̅ . As we explained in Supplementary Note 2, the fundamental group 𝑃̅ for six-band Hamiltonians has an\n𝑵 6\n8-dimensional matrix representation which can be obtained using the Dirac matrices of 6-dimensional space.\nThis 8-dimensional matrix representation forms the concrete expression of the non-Abelian charges that can\nbe computed numerically.\nIn the revised manuscript, we have added the definition of topological charge in the section of “Minimal\nnon-Abelian topological insulator model with six bands” in the main text and the Supplementary Note 3\nof the Supplementary Information.\nParagraph 3 of the section “Minimal non-Abelian topological insulator model with six bands”:\n“Considering all the bandgaps between adjacent bands, the order-parameter space of the Hamiltonian is\n𝑀 =𝑂(6)/𝑂(1)6, where 𝑂(6) is the 6D orthogonal group and 𝑂(1)6 ≅ℤ6 arises from the ±1 gauge\n6 2\nfreedom of the eigenvectors. The corresponding fundamental homotopy group is given by the Salingaros vee\ngroup1 that 𝜋 (𝑀 )=𝑃̅ =⋃ {±𝑒 𝑛1𝑒 𝑛2⋯𝑒 𝑛5}, which is formed by products of the basis elements\n1 6 6 𝑛𝑖∈{0,1} 1 2 5\nof the real Clifford algebra 𝐶ℓ : {𝑒 ,𝑒 ,𝑒 ,𝑒 ,𝑒 }, where {𝑒 ,𝑒 }=−2𝛿 .”\n0,5 1 2 3 4 5 𝑖 𝑗 𝑖𝑗\n10\n“The topological phase classification is determined by this non-Abelian fundamental group 𝑃̅ , and the\n6\ngroup elements are referred to as the non-Abelian topological charges characterizing the topological\nphases.”\nParagraph 3 of the Supplementary Note 3 “Minimal four-band non-Abelian topological insulator\nmodel”: “Considering all the gaps between neighboring bands, the order-parameter space of the gapped\nHamiltonian is 𝑀 =𝑂(4)/𝑂(1)4, where 𝑂(4) is the 4D orthogonal group and 𝑂(1)4 ≅ℤ4 arises from\n4 2\nthe ±1 gauge freedom of the eigenvectors. The corresponding fundamental homotopy group is given by the\nSalingaros vee group that 𝜋 (𝑀 )=𝑃̅ =⋃ {±𝑒 𝑛1𝑒 𝑛2𝑒 𝑛3}, where 𝑒 , 𝑒 , 𝑒 are the basis of real\n1 4 4 𝑛𝑖∈{0,1} 1 2 3 1 2 3\nClifford algebra 𝐶ℓ , satisfying {𝑒 ,𝑒 }=−2𝛿 .”\n0,3 𝑖 𝑗 𝑖𝑗\nComment 3.4:\n4. What is the relationship between the elements of group Q64 and group Q16 and the topological edge\nstates? How to analyze this connection?\nResponse:\nWe thank the reviewer for this question. The Salingaros vee groups 𝑃̅ =⋃ {±𝑒 𝑛1𝑒 𝑛2𝑒 𝑛3} (𝑄 in the\n4 𝑛𝑖∈{0,1} 1 2 3 16\nprevious notation) and 𝑃̅ =⋃ {±𝑒 𝑛1𝑒 𝑛2𝑒 𝑛3𝑒 𝑛4𝑒 𝑛 5} (𝑄 in the previous notation) are identified with\n6 𝑛𝑖∈{0,1} 1 2 3 4 5 64\nthe real Clifford algebra 𝐶ℓ and 𝐶ℓ , which are associated with the topological structures of four-band\n0,3 0,5\nand six-band Hamiltonians, respectively2-4. Therefore, the 𝑃̅ can be viewed as a subgroup of the 𝑃̅ . We can\n4 6\nunderstand the difference between them from the perspectives of eigenvector rotation as detailed in the\nfollowing.\nWe first illustrate eigenvector rotation using a three-band example in Fig. R2. To explain the role of the\nfundamental group in this context, it is helpful to use a real-space analog which is more intuitive. The\ntopology of defects in a biaxial liquid crystal can be described using exactly the same framework if we\nassociate a disclination (line defect) in the real space of a biaxial liquid crystal with a nodal-degeneracy line\nin the band structure (momentum space) of a three-band periodic system with PT symmetry. In this mapping,\nthe orientation of the ellipsoidal liquid-crystal molecular frame in real space corresponds to the orientation\nof the eigenvector frame formed by the three real eigenvectors of the k-dependent Hamiltonian in momentum\nspace. One can then consider a closed loop encircling a line defect: the orientation of the molecular frame\nafter going through one loop must come back to itself, and the rotation of along this loop provides a\ntopological classification of the enclosed disclination. By exact analogy, the rotation of the eigenvector frame\nof 𝐻(𝐤)along a loop encircling a nodal line yields a topological classification of the nodal line itself.\nThe eigenvector frame of 𝐻(𝐤) can be represented by the principal axis of biaxial liquid crystal\nmolecules. (middle inset of Fig. R2), with the blue, green, and red axes corresponding to the first, second,\nand third bands, respectively. In the left panel of Fig. R2a, liquid crystal molecules form a disclination defect\nstructure with topological charge +𝑖. The meaning of this charge is that by moving along the black circular\npath, the liquid crystal molecule (eigenvector) rotates +𝜋 around the blue axis. Charges ±𝑗 and ±𝑘 indicate\n±𝜋 rotations around the green and red axes (left panels of Figs. R2b and R2c). A charge of −1 corresponds\nto a ±2𝜋 rotation (left panel of Fig. R2d), while +1 indicates no rotation, known as \"non-Abelian frame\ncharges\". These charges can also be understood through band degeneracy. The black circle in the right panel\nof Fig. R2a surrounds the degenerate points of the second and third bands. Moving along this path from a\nregion with charge +𝑖 causes the bands to open, close, and reopen. Passing through charges +𝑗 and +𝑘 leads\nto degeneracies between the first and third bands and between the first and second bands (right panels of\nin the right panel of Fig. R2d.\nFig. R2| Understanding non-Abelian topological charge through liquid crystal defects and band\ndegeneracies. a-d Defect structures and band structures corresponding to charges +𝑖 (a), +𝑗 (b), +𝑘 (c) and\n−1 (d).\nThen we consider a flattened four-band non-Abelian Hamiltonian, 𝐻(𝑘)=𝑅(𝑘)𝐼 𝑅(𝑘)𝑇, where\n1234\n𝑅(𝑘)∈𝑆𝑂(4) is the 4D special orthogonal group, 𝑘 ∈[−𝜋,𝜋] is the Bloch wave vector and 𝐼 =\n1234\n𝑑𝑖𝑎𝑔(1,2,3,4). The Hamiltonian has four real eigenvectors as 𝐻(𝑘)|𝑛⟩=𝑛|𝑛⟩ with 𝑛 =1, 2, 3 and 4. When\n𝑘 runs from −𝜋 to 𝜋, the rotation matrix 𝑅(𝑘) will continuously act on the eigenvector |𝑛⟩ and one finally\nobtains +|𝑛⟩ or −|𝑛⟩ corresponding to the Zak phase of 0 or 𝜋, respectively. However, the Zak phase is\ninsufficient for describing eigenvector rotation, as it lacks information about the rotation planes. This\ncontrasts with the case of three-band Hamiltonians, where the quaternions ±𝑖,±𝑗,±𝑘 align with the\ndescription provided by the Zak phase. In contrast, when dealing with Hamiltonians that have more than\nthree bands, the situation changes significantly. Now there are more than one rotation plane in the parameter\nspace that the Zak phase only cannot determine the eigenvector rotation in a higher-dimensional space. For\nthe four-dimensional rotation, there exist two orthogonal rotation planes and we can set 𝑅(𝑘)=\nexp(𝜙 (𝑘)∑ 𝑛 (𝑘)𝐿 +𝜙 (𝑘)∑ 𝑛 (𝑘)𝐿 ). Here,𝜙 (𝑘) is the rotation angle and\n1 𝑖<𝑗=1:4 1,𝑖𝑗 𝑖𝑗 2 𝑖<𝑗=1:4 2,𝑖𝑗 𝑖𝑗 1(2)\nn (𝑘) determines the rotation plane. If one of the rotation planes is fixed such that 𝜙 (𝑘)=0, then\n1(2),𝑖𝑗 2\n𝑅(𝑘) corresponds to a single rotation where only the sub-eigenvector on one rotation plane rotates. Based\non the orientation of the rotation plane and the rotation angle, the following cases can be distinguished. If\nthe rotation plane is 𝐿 and the rotation angle is ±𝜋, 𝑅(𝑘) is 𝑅(𝑘)=exp[(±𝑘+𝜋)/2∙𝐿 ] and the\n𝑖𝑗 𝑖𝑗\ncorresponding non-Abelian charge is ±𝑞 such as {±𝑞 ,±𝑞 ,±𝑞 ,±𝑞 ,±𝑞 ,±𝑞 }. It indicates the\n𝑖𝑗 12 13 14 23 24 34\neigenvector rotation in the plane spanned by the eigenvectors of the 𝑖-th and 𝑗-th bands.\nTo illustrate the edge state distribution corresponding to non-Abelian charges, we introduce the concept\nof the extended 2D plane. If we make the substitutions like cos𝑘 →𝑘 and sin𝑘 →𝑘 to expand the 1D\n1 2\nHamiltonian 𝐻(𝑘) to a 2D Hamiltonian 𝐻(𝑘 ,𝑘 ), we can get the extended 2D band structures as shown in\n1 2\nthe left panels of Figs. R3a-3b. The original 1D Hamiltonian 𝐻(𝑘) corresponds to a unit circle in this 𝑘 −\n1\n𝑘 plane. In Fig. R3a, one linear degeneracy can be observed between the 1st and 2nd bands for the charge\n2\n12\n±𝑞 . As the loop defining the 1D Hamiltonian is moving in parameter space and crosses the degeneracy\n12\npoint, the energy bands will close and then reopen. According to the Jackiw-Rebbi argument1, edge states\nwill emerge around the gap closing point. This suggests that edge states emerge in the gap between the 1st\nand 2nd bands, as confirmed by the eigenstate patterns of a finite lattice with 21 unit cells, shown in the right\npanel of Fig. R3a. The horizontal and vertical axes represent the real-space locations and the indices of the\neigenstates, respectively. The inset on the right displays the energy levels, with red and black dots indicating\nbulk and edge states, respectively. Similarly, we calculate the extended 2D band and the eigenstate patterns\nfor the charge ±𝑞 as shown in Fig. R3b. A linear degeneracy is observed between the 1st and 3rd bands,\n13\nindicating the emergence of a pair of edge states between them, which accidentally emerge within the 2nd\nband. These results suggest that the degenerate points of the extended 2D plane can, to some extent, reflect\nthe distribution of edge states. Therefore, the extended 2D plane can often be used to roughly explore the\ndistribution of edge states. However, it is important to note that the degenerate points of the extended 2D\nplane and edge states do not form a strict one-to-one correspondence. This is because the construction of the\nextended 2D plane is not unique, which is especially evident in cases where the charge is composed of\nmultiple charge combinations, such as ±𝑞 and −1. Therefore, we can only use this approach for a\n1234\nheuristic understanding.\nFig. R3| Extended 2D band structures and edge states. a-b Extended 2D band structures and edge states\nof charges ±𝑞 (a) and ±𝑞 (b).\n12 13\nIn Fig. R4a, we show the extended 2D band structures of ±𝑞 , ±𝑞 , ±𝑞 , ±𝑞 , ±𝑞 , ±𝑞\n12 13 14 23 24 34\nrespectively and linear degeneracies can be observed between the 𝑖-th and 𝑗-th bands, indicating the edge\nstates show up in the gap between the 𝑖-th and 𝑗-th bands. If the rotation angle is 2𝜋, 𝑅(𝑘) is 𝑅(𝑘)=\nexp[(𝑘+𝜋)𝐿 ] and the corresponding non-Abelian charge is −1. The extended 2D band structure of −1\n𝑖𝑗\ncontains quadratic degeneracies as shown in the upper panel of Fig. R4b, indicating that at least two pairs of\nedge states exist. The non-Abelian charge +1 corresponds to a trivial rotation, meaning there are no\ndegeneracies or topological edge states observed (lower panel of Fig. R4b).\n12 13 14 23 24 34\n+1 (b), respectively.\nThe above non-Abelian charges {±𝑞 ,±𝑞 ,±𝑞 ,±𝑞 ,±𝑞 ,±𝑞 }∪{+1,−1} can find their\n12 13 14 23 24 34\ncounterparts in three-band models, as their eigenvector rotations are single rotations. However, for the non-\nAbelian charges ±𝑞 involving two orthogonal rotation planes that 𝜙 (𝑘)≠0 and 𝜙 (𝑘)≠0 , they do\n1234 1 2\nnot have their counterparts in the three-band model. According to the chosen of the two orthogonal rotation\nplanes, the charge ±𝑞 can be factorized as ±𝑞 𝑞 , ∓𝑞 𝑞 and ±𝑞 𝑞 . For example, the\n1234 12 34 13 24 14 23\nfactorization 𝑞 =𝑞 𝑞 corresponds to 𝑅(𝑘)=exp[(𝑘+𝜋)/2∙𝐿 +(𝑘+𝜋)/2∙𝐿 ], where the\n1234 12 34 12 34\ntwo rotation planes are spanned by the 1-2 bands and 3-4 bands respectively. We also show the corresponding\nextended 2D band degeneracies of the three cases as shown in Figs. R5a-c. For the three factorizations, the\nunit circles encircle different degeneracies, indicating different edge state distributions. Note that they belong\nto the same topological charge and can be continuously transformed into each other without closing the\nbandgap at the unit circle. Therefore, the distribution of edge states for ±𝑞 remains undetermined and\n1234\ncomplex, and a precise prediction is still an open question.\nFig. R5| Extended 2D band structures of charges corresponding to double rotations in group 𝑷̅ . a-c\n12 34 13 24 14 23\nrespectively.\nedge state distribution, we will further extend the discussion to the six-band model. In the same way, we also\nconstruct a flattened six-band non-Abelian Hamiltonian as 𝐻(𝑘)=𝑅(𝑘)𝐼 𝑅(𝑘)𝑇, where 𝐼 =\n123456 123456\n𝑑𝑖𝑎𝑔(1,2,3,4,5,6) and 𝑅(𝑘)=exp(𝜙 (𝑘)∑ 𝑛 (𝑘)𝐿 +𝜙 (𝑘)∑ 𝑛 (𝑘)𝐿 +\n1 𝑖<𝑗=1:6 1,𝑖𝑗 𝑖𝑗 2 𝑖<𝑗=1:6 2,𝑖𝑗 𝑖𝑗\n𝜙 (𝑘)∑ 𝑛 (𝑘)𝐿 ). If one of the rotation planes is fixed such that 𝜙 (𝑘)=0, the situation\n3 𝑖<𝑗=1:6 3,𝑖𝑗 𝑖𝑗 3\ndegenerates to that of the four-band model. Therefore, the 𝑃̅ of the four-band system can be regarded as a\n4\nsubgroup of the 𝑃̅ of the six-band system. The left panel of Fig. R6a shows an example of an extended 2D\n6\nband associated with ±𝑞 in a 𝑃̅ flat-band model. In this configuration, linear degeneracies occur\n1234 6\nbetween the energy gaps 1-3 and 2-4. This observation aligns with the presence of edge states, represented\nby red dots, in the energy levels of the corresponding 1D finite lattice composed of 10 unit cells, as depicted\nin the right panel of Fig. R6a. Compared with 𝑃̅ , 𝑃̅ contains a special class of topological charges ±𝑞 ,\n4 6 123456\nwhich correspond to triple rotations with three sets of orthogonal rotation planes. This arises from the fact\nthat their eigenstate rotations occur in a six-dimensional rotational space. Similar with the charges ±𝑞\n1234\nin the four-band model, the charges ±𝑞 can be factorized into several simple charges. For example,\n123456\nthe factorization ±𝑞 = ±𝑞 𝑞 𝑞 indicates that the rotation planes correspond to the 1-4, 2-5,\n123456 14 25 36\nand 3-6 bands, with edge states residing in the respective gaps: 1-4, 2-5, and 3-6. This is verified by the\nextended 2D band and the energy levels as shown in Fig. R6b. There are a total of 𝐶2𝐶2⁄3!= 15\n6 4\nfactorizations, resulting in more intricate distributions of edge states.\nFig. R6| Extended 2D band structures and edge states of charges in group 𝑷̅ . a-b Extended 2D band\n𝟔\nstructures (left panels) and the corresponding energy levels associated with the 1D lattice of 10 unit cells\n(right panels) for the factorizations ±𝑞 𝑞 (a) and ±𝑞 𝑞 𝑞 (b), respectively.\n13 24 14 25 36\nIn the revised Supplementary Information, we have added a new section (Supplementary Note 6)\ndescribing the extended 2D band for six-band models titled “Extended 2D planes for the six-band model”\nand included Fig. R6 as Supplementary Fig. 9.\nSupplementary Note 6 “Extended 2D planes for the six-band model”: “To understand the non-Abelian\ntopological charge of six bands from a higher-dimensional perspective, we will introduce the concept of the\nextended 2D plane. We consider a flattened six-band non-Abelian Hamiltonian as 𝐻(𝑘)=\n𝑅(𝑘)𝐼 𝑅(𝑘)𝑇, where 𝐼 =𝑑𝑖𝑎𝑔(1,2,3,4,5,6), 𝑘 ∈[−𝜋,𝜋] is the Bloch wave vector and 𝑅(𝑘)∈\n123456 123456\n𝑆𝑂(6) is the 6D special orthogonal group that 𝑅(𝑘)=𝑒𝑥𝑝(𝜙 (𝑘)∑ 𝑛 (𝑘)𝐿 +\n1 𝑖<𝑗=1:6 1,𝑖𝑗 𝑖𝑗\n𝜙 (𝑘)∑ 𝑛 (𝑘)𝐿 +𝜙 (𝑘)∑ 𝑛 (𝑘)𝐿 ). This Hamiltonian has six real eigenvectors as\n2 𝑖<𝑗=1:6 2,𝑖𝑗 𝑖𝑗 3 𝑖<𝑗=1:6 3,𝑖𝑗 𝑖𝑗\n𝐻(𝑘)|𝑛⟩=𝑛|𝑛⟩ with 𝑛 =1, 2, 3, 4, 5 and 6. When 𝑘 runs from −𝜋 to 𝜋, the rotation matrix 𝑅(𝑘) will\ncontinuously act on the eigenvector |𝑛⟩ and one finally obtains +|𝑛⟩ or −|𝑛⟩ corresponding to the Zak\nphase of 0 or 𝜋, respectively. If one of the rotation planes is fixed such that 𝜙 (𝑘)=0, the situation\n3\n15\ndegenerates to that of the four-band model. Therefore, the 𝑃̅ of the four-band system can be regarded as a\n4\nsubgroup of the 𝑃̅ of the six-band system.\n1 2\n2D Hamiltonian 𝐻(𝑘 ,𝑘 ), we can get the extended 2D band structures as shown in Supplementary Fig. 9.\n1 2\nThe original 1D Hamiltonian 𝐻(𝑘) corresponds to a unit circle in this 𝑘 −𝑘 plane. The left panel of\n1 2\nSupplementary Fig. 9a shows an example of an extended 2D band corresponding to ±𝑞 in a 𝑃̅ flat-\n1234 6\nband model. One linear degeneracy is observed between the 1-3 bands, while another occurs between the\n2-4 bands. This indicates the factorization of ±𝑞 as ±𝑞 𝑞 . As the loop defining the 1D Hamiltonian\n1234 13 24\nis moving in parameter space and crosses the degeneracy point, the energy bands will close and then reopen.\nAccording to the Jackiw-Rebbi argument1, edge states will emerge around the gap closing point. This\nsuggests that edge states emerge in the gap between the 1-3 bands and 2-4 bands, which is verified by the\nenergy levels of the corresponding 1D finite lattice of 10 unit cells as shown in the right panel of\nSupplementary Fig. 9a. Compared with 𝑃̅ , 𝑃̅ contains a special class of topological charges ±𝑞 ,\n4 6 123456\nwhich correspond to triple rotations with three sets of orthogonal rotation planes. This arises from the fact\nthat their eigenstate rotations occur in a six-dimensional rotational space. Similar with the charges ±𝑞\n1234\nin the four-band model, the charges ±𝑞 can be factorized into several simple charges. For example,\n123456\nthe factorization ±𝑞 =±𝑞 𝑞 𝑞 shows that the rotation planes correspond to the 1-4, 2-5, and 3-\n123456 14 25 36\n6 bands, with edge states residing in the respective gaps: 1-4, 2-5, and 3-6. This is verified by the extended\n2D band and the energy levels as shown in Supplementary Figure 9b. There are a total of 𝐶2𝐶2⁄3!=15\n6 4\nfactorizations, resulting in more intricate distributions of edge states.”\nComment 3.5:\n5. Although the supplementary materials present the research results of the four-band system, they do not\nsystematically compare the commonalities and differences between the four-band and six-band systems in\nterms of topological charge classification, phase transition rules, and edge state characteristics, such as\nwhether an increase in the number of bands will introduce new topological phases, which fails to reflect the\nregularity of model expansion.\nResponse:\nWe thank the reviewer for this question. As discussed in our reply to Comment 3.4, 𝑃̅ in the four-band\n4\nsystem is a subgroup of 𝑃̅ in the six-band system, with the former corresponding to 4D rotation and the\n6\nlatter to 6D rotation. The 𝑃̅ group has an additional special topological charge ±𝑞 compared to 𝑃̅ ,\n6 123456 4\nwhich arises from the unique triple rotation in 6D rotations, a feature that has no counterpart in 4D rotations.\nGenerally, for a multi-band system (𝑁 ≥ 3), its fundamental homotopy group is given by the real\nClifford algebra 𝐶ℓ , and is the Salingaros vee group 𝑃̅ =⋃ {±𝑒 𝑛1𝑒 𝑛2…𝑒 𝑛𝑁−1}. Since the\n0,𝑁−1 𝑁 𝑛𝑖∈{0,1} 1 2 𝑁−1\nvalues of 𝑛 can only be 0 or 1, with the distinction of signs, the number of elements in the group 𝑃̅ is 2𝑁.\n𝑖 𝑁\nAs the number of bands increases, so does the number of orthogonal rotation planes, leading to the addition\nof new special topological charges on top of the existing subgroups. For instance, when the number of bands\nis increased to eight, the eigenstate rotation becomes an 8D rotation, involving up to four orthogonal rotation\nplanes. If all four planes rotate by ±𝜋, the corresponding additional special topological charges are\n±𝑞 . However, if the number of bands is increased to seven, the eigenstate rotation becomes a 7D\n12345678\nrotation, involving only three orthogonal rotation planes, with the remaining dimension (7 − 2∗3 = 1)\nnot participating in the rotation. Removing this extra dimension, the 7D rotation degenerates into a 6D\n16\nrotation, so the seven-band system does not introduce a new special topological charge compared to the six-\nband system.\nIt is important to note that the introduction of this extra dimension is not without significance. In the\nbased homotopy classification, generally, two elements that differ only by a sign, such as +𝑒\n𝑛1𝑒 𝑛2…𝑒 𝑛𝑁−1\n1 2 𝑁−1\nand −𝑒\n𝑛1𝑒 𝑛2…𝑒 𝑛𝑁−1,\nbelong to the same conjugacy class, except for the cases of +1 and −1. However,\n1 2 𝑁−1\nthis does not apply to topological charges for all bands that have non-trivial Zak phases. In the six-band\nsystem, +𝑞 and −𝑞 belong to two distinct conjugacy classes, as they cannot be transformed into\n123456 123456\none another by conjugation. However, in the seven-band system, they belong to the same conjugacy class\nbecause the additional dimension provides an extra conjugation operation, making it possible for one to be\nconjugated into the other: 𝑞 𝑞 𝑞−1 =−𝑞 .\n67 123456 67 123456\nTherefore, for an odd-numbered (𝑁 =2𝑛−1) band system, the eigenstate rotation is a (2𝑛−1)-\ndimensional rotation, with the number of orthogonal rotation planes being 𝑛−1, and the number of\nconjugacy classes being 2𝑁−1+1. For an even-numbered (𝑁 =2𝑛) band system, the eigenstate rotation is\na 2𝑛- dimensional rotation, with the number of orthogonal rotation planes being 𝑛, and the number of\nconjugacy classes being 2𝑁−1+2. This rule is summarized in Table R1 below.\nOdd number (𝑁 =2𝑛−1,𝑛 ≥\nBand number Even number (𝑁 =2𝑛,𝑛 ≥2)\n2)\nRotating dimensions (2𝑛−1)-dimensional 2𝑛-dimensional\nNumber of orthogonal\n𝑛−1 𝑛\nrotation planes\nNumber of conjugacy classes 2𝑁−1+1 2𝑁−1+2\nNumber of non-Abelian charges 2𝑁 2𝑁\nTable R1| The difference in topological classification between odd and even bands.\nIn the revised Supplementary Information, we have added a new section (Supplementary Note 7)\ndescribing the cases of arbitrary band numbers titled “Generalized quaternion charge for systems with\narbitrary band numbers” and included Table R1 as Supplementary Table 9.\nSupplementary Note 7 “Generalized quaternion charge for systems with arbitrary band numbers”:\nIn this section, we examine systems with an arbitrary number of bands. Generally, for a multi-band system\n(𝑁 ≥ 3), its fundamental homotopy group is given by the real Clifford algebra 𝐶ℓ , and is the\n0,𝑁−1\nSalingaros vee group 𝑃̅ =⋃ {±𝑒 𝑛1𝑒 𝑛2…𝑒 𝑛𝑁−1}. Since the values of 𝑛 can only be 0 or 1, with the\n𝑁 𝑛𝑖∈{0,1} 1 2 𝑁−1 𝑖\ndistinction of signs, the number of elements in the group 𝑃̅ is 2𝑁. As the number of bands increases, so does\n𝑁\nthe number of orthogonal rotation planes, leading to the addition of new special topological charges on top\nof the existing subgroups. For instance, when the number of bands is increased to eight, the eigenstate\nrotation becomes an 8D rotation, involving up to four orthogonal rotation planes. If all four planes rotate\nby ±𝜋, the corresponding additional special topological charges are ±𝑞 . However, if the number\n12345678\nof bands is increased to seven, the eigenstate rotation becomes a 7D rotation, involving only three\northogonal rotation planes, with the remaining dimension (7 − 2∗3 = 1) not participating in the rotation.\nRemoving this extra dimension, the 7D rotation degenerates into a 6D rotation, so the seven-band system\ndoes not introduce a new special topological charge compared to the six-band system.\nbased homotopy classification, generally, two elements that differ only by a sign, such as +𝑒\n𝑛1𝑒 𝑛2…𝑒 𝑛𝑁−1\n1 2 𝑁−1\nand −𝑒\n𝑛1𝑒 𝑛2…𝑒 𝑛𝑁−1,\nbelong to the same conjugacy class, except for the cases of +1 and −1. However,\n1 2 𝑁−1\nthis does not apply to topological charges for all bands that have non-trivial Zak phases. In the six-band\nsystem, +𝑞 and −𝑞 belong to two distinct conjugacy classes, as they cannot be transformed into\n123456 123456\none another by conjugation. However, in the seven-band system, they belong to the same conjugacy class\nbecause the additional dimension provides an extra conjugation operation, making it possible for one to be\nconjugated into the other: 𝑞 𝑞 𝑞−1 =−𝑞 .\n67 123456 67 123456\nTherefore, for an odd-numbered (𝑁 =2𝑛−1) band system, the eigenstate rotation is a (2𝑛−1)-\ndimensional rotation, with the number of orthogonal rotation planes being 𝑛−1, and the number of\nconjugacy classes being 2𝑁−1+1. For an even-numbered (𝑁 =2𝑛) band system, the eigenstate rotation is\na 2𝑛- dimensional rotation, with the number of orthogonal rotation planes being 𝑛, and the number of\nconjugacy classes being 2𝑁−1+2. This rule is summarized in Supplementary Table 9.\nComment 3.6:\n6. The article mentions the distinction between bulk states and edge states through the inverse participation\nratio (IPR), but does not provide a specific definition of IPR (nor does it explain the selection criteria for the\nIPR threshold), resulting in an unclear basis for distinguishing marginal states from postures).\nResponse:\nWe appreciate the reviewer for this suggestion. For a normalized eigenstate, its IPR is defined as IPR=\n∑ |𝜑 |4, where the 𝜑 satisfy the normalization condition ∑ |𝜑 |2 = 1. The IPR provides information about\n𝑖 𝑖 𝑖 𝑖 𝑖\nhow many lattice sites the eigenstate “occupies”. If a state is completely localized at a single lattice site |𝑘⟩,\ni.e., 𝜑 = 1 and all other 𝜑 = 0, then IPR= ∑ |𝜑 |4 = |𝜑 |4 = 1, which is the maximum value of the\n𝑘 𝑖 𝑖 𝑖 𝑘\nIPR. Conversely, if a state is fully extended, with its wave function uniformly distributed over 𝑁 lattice sites,\nthen as the system size 𝑁 →∞, the IPR approaches 0. This represents the minimum value of the IPR for an\ninfinitely large system. In Fig. 1g of the main text, states with IPR greater than 0.15 are considered boundary\nstates and marked in red, while states with IPR less than 0.15 are regarded as bulk states and marked in gray.\nThe maximum and minimum values of the IPR are 0.4846 and 0.0095, respectively. In Supplementary Fig.\n1f of the Supplementary Information, states with IPR greater than 0.13 are considered boundary states and\nmarked in red, while states with IPR less than 0.13 are regarded as bulk states and marked in gray. The\nmaximum and minimum values of the IPR are 0.4615 and 0.0186, respectively.\nIn the revised manuscript, we have included this description of the IPR in the section of “Edge/domain-\nwall states of six-band non-Abelian topological insulators” in the main text and the Supplementary Note\n3 in the Supplementary Information. We have also revised the tick labels of the color bar in Fig. 1g of the\nmain text and Supplementary Fig. 1f of the Supplementary Information.\nParagraph 1 of the section “Edge/domain-wall states of six-band non-Abelian topological\ninsulators”: “The IPR is defined as IPR=∑|𝜑|4, where 𝜑 is the normalized wave function at lattice 𝑖.\n𝑖 𝑖 𝑖\nThe states with a large IPR (appear in red) are considered edge states, while those with a small IPR (appear\nin gray) are considered bulk states. The corresponding IPR threshold is approximately 0.15.”\nParagraph 5 of the Supplementary Note 3 “Minimal four-band non-Abelian topological insulator\nmodel”: “The states with a large IPR (appear in red) are considered edge states, while those with a small\nIPR (appear in gray) are considered bulk states. The corresponding IPR threshold is approximately 0.13.”\n7. The experiment observes the phase transition by changing the gap $g_{out}$ at the output end of the\nsample. Essentially, it is a static comparative experiment rather than a real-time dynamic observation, which\ncannot capture the evolution dynamics of the topological state during the phase transition process (such as\nthe time scale of edge mode disappearance), thus limiting the in-depth understanding of the non-Abelian\nphase transition mechanism. This also goes against the authors' original intention, as in the introduction, the\nauthors explicitly mentioned the study of dynamical phase transitions.\nResponse:\nWe appreciate the comment by the reviewer. The reason we claim our system as a “dynamical” system is\nthat light propagating in the paraxial waveguides is governed by 𝐻(𝑧)|𝜑(𝑧)⟩=−𝑖𝜕 |𝜑(𝑧)⟩, where 𝐻(𝑧)\n𝑧\nand |𝜑(𝑧)⟩ are the Hamiltonian and state vector, respectively. The form of the master equation is in fact a\ntime-dependent Schrödinger-like equation in which the original time term t is replaced by a space dimension\nterm z. In our experiment, as the gap distance (which determines the topological charge) varies along the z\naxis, light propagating in the system would experience multiple non-Abelian topological phases at different\n“timeslots” 𝑧. Since different topological phases support different distributions of edge states, measuring the\nexistence of edge states in such systems would help us identify different topological phases and the phase\ntransition between them. This is motivation that we design such a system with varying structure parameters.\nWe agree with the reviewer that our experiment is a static comparative setup rather than a dynamic one.\nAlthough the phase transitions can be probed in numerical simulations such as those in Fig. 3d,e, they cannot\nbe directly observed in experiments, since phase transitions only occur during the light evolution process,\nwhile we can only measure the light distributions at the output side. Therefore, claiming that we have\nobserved the phase transition in experiment is not strict.\nIn the revised manuscript, we have removed all the statements associated with “dynamical phase\ntransition” in the Abstract, main text and Supplementary Information. Instead, we claim that the phase\ntransition can be probed in numerical simulations in the system with varying structural parameters, while in\nexperiment, we only identify different topological phases. We have rewritten all the corresponding contents.\nSome key revisions are listed below for the reviewer to check.\nAbstract: “Different non-Abelian topological charges have been experimentally demonstrated by probing\nthe adiabatic evolution of edge states in a system with varying structural parameters.”\nParagraph 2 in Introduction: “We design and implement an experimental configuration in which light\ncould encounter structures with different non-Abelian topological phases along the propagation direction.\nIn this way, different non-Abelian topological charges can be revealed in experiments.”\nParagraph 3 of the section “Experimental demonstration of edge states associated with different non-\nAbelian topological charge”: “In experiments, the phase transition cannot be directly observed, as it occurs\nduring the light evolution process. Instead, this setting can be employed to distinguish different topological\nphases, as crossing the phase transition point could affect the feature of the excited edge states.”\nComment 3.8:\n8. What is the motivation for choosing s=t to conduct research? What changes will occur in the result if s is\ndifferent from t?\nResponse:\nsetting 𝑠 =𝑡, except for convenience. When 𝑠 ≠𝑡, as long as the deviation is not large, the corresponding\nphase diagram does not show any significant difference. As shown in Fig. R7, we have calculated two phase\ndiagrams corresponding to 𝑡 =0.8𝑠 =0.8𝛥 and 𝑡 =1.2𝑠=1.2𝛥 , with the other parameters unchanged,\nconsistent with Fig. 1g in the main text. It can be observed that the phase transition points shift under different\nparameters, but the overall difference in the phase diagram is not substantial.\nIn the revised Supplementary Information, we have included Fig. R7 as Supplementary Fig. 10.\nFig. R7| Phase diagrams for 𝒔≠𝒕. a-b Phase diagrams corresponding to 𝑡 =0.8𝑠 =0.8𝛥 (a) and 𝑡 =\n1.2𝑠=1.2𝛥 (b), respectively. The other parameters are consistent with Fig. 1g in the main text.\nReferences\n1. Jackiw, R. & Rebbi, C. Solitons with fermion number ½. Phys. Rev. D 13, 3398 (1976).\n2. Wu, Q., Soluyanov, A. A. & Bzdušek, T. Non-Abelian band topology in noninteracting metals. Science\n365, 1273-1277 (2019).\n3. Guo, Q. et al. Experimental observation of non-Abelian topological charges and edge states. Nature 594,\n195-200 (2021).\n4. Jiang, T. et al. Four-band non-Abelian topological insulator and its experimental realization. Nat.\nCommun. 12, 6471 (2021).\nI am satisfied with the response and updates. I would like to urge the authors to add\n[1] J. Kruthoff, J. de Boer, J. van Wezel, C. L. Kane, and R.J. Slager, Topological classification of crystalline\ninsulators through band structure combinatorics, Phys. Rev. X 7, 041069 (2017).\n[2] H. C. Po, A. Vishwanath, and H. Watanabe, Symmetry- based indicators of band topology in the 230\nspace groups, Nature Communications 8, 50 (2017).\n[3] B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig,\nTopological quantum chemistry, Nature 547, 298 (2017)\nto Ref 1,2. This because the non-Abelian or multigap phases precisely go beyond these complete single gap\nvariants.\nResponse:\nWe thank the referee for his/her efforts in reviewing our work. We have added the above three references as\nRefs. 3-5 in the final version of the manuscript.\nResponse to Reviewer 3\nThe author made detailed revisions and responses to the questions raised by the reviewers, which made the\narticle's highlights and significance more prominent, and also enabled me to have a deeper understanding of\nphoton non-abelian topology. I approve of the author's revisions and responses, and I agree that this research\nshould be published in Nature Communications.\nResponse:\nWe thank the referee for his/her efforts in reviewing our work and the recommendation for publication.\n1"}, {"round": 2, "reviewer_comments": {"reviewer_1": "I am satisfied with the response and updates. I would like to urge the authors to add ]\n[1] J. Kruthoff, J. de Boer, J. van Wezel, C. L. Kane, and R.J. Slager, Topological classification of crystalline insulators\nthrough band structure combinatorics, Phys. Rev. X\n7, 041069 (2017).\n[2] H. C. Po, A. Vishwanath, and H. Watanabe, Symmetry- based indicators of band topology in the 230 space groups,\nNature Communications 8, 50 (2017).\n[3] B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Topological quantum\nchemistry, Nature 547, 298 (2017)\nto Ref 1,2. This because the non-Abelian or multigap phases precisely go beyond these complete single gap variants.", "reviewer_3": "The author made detailed revisions and responses to the questions raised by the reviewers, which made the article's\nhighlights and significance more prominent, and also enabled me to have a deeper understanding of photon non-abelian\ntopology. I approve of the author's revisions and responses, and I agree that this research should be published in Nature\nCommunications."}, "author_response": "Response to Reviewer 1\nI am satisfied with the response and updates. I would like to urge the authors to add\n[1] J. Kruthoff, J. de Boer, J. van Wezel, C. L. Kane, and R.J. Slager, Topological classification of crystalline\ninsulators through band structure combinatorics, Phys. Rev. X 7, 041069 (2017).\n[2] H. C. Po, A. Vishwanath, and H. Watanabe, Symmetry- based indicators of band topology in the 230\nspace groups, Nature Communications 8, 50 (2017).\n[3] B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig,\nTopological quantum chemistry, Nature 547, 298 (2017)\nto Ref 1,2. This because the non-Abelian or multigap phases precisely go beyond these complete single gap\nvariants.\nResponse:\nWe thank the referee for his/her efforts in reviewing our work. We have added the above three references as\nRefs. 3-5 in the final version of the manuscript.\nResponse to Reviewer 3\nThe author made detailed revisions and responses to the questions raised by the reviewers, which made the\narticle's highlights and significance more prominent, and also enabled me to have a deeper understanding of\nphoton non-abelian topology. I approve of the author's revisions and responses, and I agree that this research\nshould be published in Nature Communications.\nResponse:\nWe thank the referee for his/her efforts in reviewing our work and the recommendation for publication."}]

Decision letter