Non-Abelian charged nodal links in dielectric photonic crystal

A nodal link is a special form of a line degeneracy (a nodal line) between adjacent bands in the momentum space of a three-dimensional topological crystal. Unlike nodal chains or knots, a nodal link consists of two or more mutually-linked rings that do not touch each other. Recent studies on non-Abelian band topology revealed that the topological charges of the nodal links can have the properties of quaternions. However, a photonic crystal that has a nodal link with non-Abelian charges has not been reported. Here, we propose dielectric photonic crystals in forms of double diamond structures which realize the nodal links in the momentum space. By examining the evolution of the eigenstate correlations along the closed loops which enclose the nodal line(s) of the links, their non-Abelian topological charges are also analyzed. The proposed design scheme and theoretical approach in this work will allow for experimental observation of photonic non-Abelian charges in purely dielectric materials and facilitate the control of the degeneracy in complex photonic structures.

Despite the abundance of studies on the nodal lines 13,16,30,36,38 and the non-Abelian band topology in photonics 38 , there has not been any textbook example on the nodal link. A photonic crystal can be such an example if (i) its output nodal lines completely satisfy the aforementioned definition of the nodal link 41 and (ii) the nodal links of the photonic crystal have non-Abelian topological charges 15 . The existing photonic crystals on the nodal lines do not completely fulfil at least one of these. In particular, although the second condition has been dealt with using the effective 3×3 Hamiltonian derived from the Maxwell's equations 38 , there is not any precedent in which the non-Abelian topological charges are calculated based on the full-vector field form of eigenstates computed for a photonic crystal. Additionally, most of the existing photonic studies on the nodal line 16 , nodal ring 30 , nodal chain 36 , or nodal links 38 are commonly limited to the metallic structures with mirror symmetry.
In this work, we theoretically demonstrate the nodal link with non-Abelian charges by using a dielectric photonic crystal. First, we introduce our double diamond structure which has inversion symmetries but does not have mirror symmetry. Then, the nodal link in the threedimensional (3D) momentum space of this structure is discussed. The non-Abelian topological features of the nodal link are characterized by considering a loop that encloses one or more sections of the link and examining the correlations of the eigenstates.
The realization of the nodal link in a dielectric photonic crystal starts with taking the wellknown diamond structure 45 and subsequently breaking its several geometrical symmetries as will be explained below. For the face-centered cubic (FCC) primitive cell whose lattice vectors still mutual-inversion symmetric and each diamond is also self-inversion symmetric that the inverse of a single diamond coincides itself. Here, the self-inversion symmetric points of the two single diamonds are identical. The second modification is setting nonzero which splits the self-inversion symmetric points along 1, 1, 1 -direction. This also eliminates the translational symmetry with the unit vector 〈 2 ⁄ , 0,0〉 so that not the primitive cubic cell with 2 ⁄ but the above FCC cell has to be used (the detailed analyses on the symmetries of the double diamond structures are discussed in Supplementary Section 2). The mutual-inversion symmetry by two single diamond structures survives which is a necessary condition for the formation of a nodal link 13,35 . Each diamond structure in these configurations is depicted in Fig.   1a; the sets of which satisfies and are given as the pink and skyblue colored structures, respectively.
The 3D photonic band structure for the double diamond photonic crystal has multiple degeneracies that form a link. To probe all the degeneracies, we calculated the photonic band structure by using the MIT Photonic-Bands (MPB) package 50 . We regarded any point as degenerated by adjacent two bands if the normalized frequency difference (∆ /2 ) of these bands at the point is smaller than a critical tolerance, 0.0045. Sets of the degeneracies formed by the 3rd and 4th bands and 4th and 5th bands appear as rings, depicted as orange and cyan shapes, respectively, in The topological charges of the link can be described by the non-Abelian quaternion group 2) is then considered (Fig. 2a). We define the correlation of the band n, also known as the polarization 38  the sign of this charge may also be flipped. This is discussed in Supplementary Section S6.
Meanwhile, the topological charge in Fig. 3 also can be drawn without the above calibration and the discussions are in Supplementary Section S7.
Besides, the topological charges and can be deduced from the above results.
Selecting the same kind of nodes (i.e., same-colored nodes) and setting the closed loop which ties oppositely oriented nodes will generate . If the loop ties nodes with the same orientation, the result will be . The former case corresponds to which increases from zero to 2 followed by decreasing to zero. Applying this path to the orange ring makes and rotate by for ∈ 0, 2 and by for ∈ 2 , 0 while keeping (Fig. 4a). The net evolutions of both and are zero and therefore the topological charge becomes . The latter case is expressed by setting ∈ 0, 4 . Along the path, and rotate by 2 around as shown in Fig.   4b. This situation is simply written by , as mentioned regarding    Fig. 2. Derivation of the topological charges i and k. a and d, Closed loops which enclose the orange and cyan nodal rings, respectively. Their winding directions and starting points are also marked as arrows and circles, respectively. b and e, Correlations from the eigenstates, along the closed loops. Only their real parts are plotted. c and f, Correlations whose vector tails are collected at the origin to see their topological charges. In b-c and e-f, we set the arbitrary orthonormal directions as , , and .
x 1 Fig. 3. Derivation of the topological charge j. a, Closed loop which encloses the cyan and orange nodal rings. Its winding direction and starting point are also marked as an arrow and a circle, respectively. b and c, Correlations plotted by same method in Fig. 2. d, at an arbitrary and -axis. The matrix , is defined such that this matches onto -axis with respect to perpendicular to both and . e and f, Calibration results of c by , 0.6 and , , respectively, where is the angle between and .

Fig. 4. ±1 Topological charges. a-d, Correlations
of the eigenstates along the loops which enclose the orange (a-b) and cyan (c-d) nodes. Only the real parts of the correlations are plotted. All these four panels have two circles in the enlargement insets. The winding directions of these two circles are opposite in a and c and same in b and d.

Competing interests
The authors declare that they have no competing interests.

Additional information
Correspondence and requests for materials should be addressed to S.S.O.

Section 2. Symmetries of the double diamond structure
Equation (1)  Their topological nature can be determined by using the same method mentioned in the main text. At first, a closed loop is considered which encircles a section in the nodal line ( Fig. S2ab). Along this loop, the correlation vectors are calculated. Gathering all tails of these vectors generates Fig. S2c. Therefore, these nodal lines are also topologically nontrivial and their topological nature can be considered as non-Abelian charge, .  x 1 Fig. S3. a, Summary of the nodal link and the chain by our double diamond photonic crystal. The pink nodal chain, cyan and orange rings are formed by the 1st and 2nd bands, 3rd and 4th bands, and 4th and 5th bands in the band structure, respectively. b, Plot of only nodal chain. c, Plot of only nodal link discussed throughout the main text.

Section 5. Sign convention of the eigenstates
The correlation whose components are defined by is the main concept which explains the non-Abelian quaternion charges throughout this study.
However, there is an important preparation to calculate the correlations. Basically, all eigenstates obtained from an eigenvalue problem exhibit the sign ambiguity, i.e., if a state | ⟩ is an eigenstate of a Hamiltonian at , | ⟩ is also the eigenstate. Thus, if we know | ⟩ with its correct sign, the sign of | ∆ ⟩ can be determined such that Re⟨ | ∆ ⟩ 0.
The problem is how to guarantee that the signs of | at the starting points of the several To elucidate this ambiguity, we use a common reference point (also known as the base point 1,2 ) and lines which connect and of each loop as shown in Fig. S4. When the topological charge on the orange path in Fig. S4 is investigated, | is considered above all.
First, the sign of | ∆ is determined such that satisfies the criterion Re Same convention is applied to the cyan path in Fig. S4.
The difference of the reference point method, compared to Refs. 1,2, is that this study does not need the reverse line from to because the correlation is defined with respect to not | but | and the aim of the reference point and the line in this study is only determining the signs of | with the same convention.
The correlation is calculated from the sign-corrected | according to this sign convention. All discussions on the non-Abelian topological charges in this study were done based on this preparation. All the preparations use a common . All closed loops shown in figures in this study omit the lines between and and all plots on the correlations show for only after .

Section 6. Discussions on signs of the topological charge j
From the closed loop shown in Fig. 3a in the main text, we mentioned its topological charge is . This is consistent with the relation due to the first and second half of the loop circle around cyan and orange nodes, respectively.
Then, we questioned about whether the topological charge is flipped from to by changing this scanning sequence. If so, the topological charge strongly reflects the non-Abelian nature of the topological charges.
The first variation on the loop is scanning around the orange node followed by the cyan node without flipping the winding direction (Fig. S5a). Keeping the winding direction means the signs of and remain. The scanning sequence makes a composite relation so we can expect its topological charge is . Indeed, the simulated correlations and reveal the rotations along the opposite direction which corresponds to (Fig. S5b).
The second variation on the loop is same to the original loop in Fig. 3a in the main text except the opposite winding direction (Fig. S5c). Due to this winding, the topological charges around the orange and cyan nodes are and , respectively. Their composition is expressed as , thus . The correlations in Fig. S5d also match to this expectation.
The final variation on the loop envelops the cyan node first and its winding direction is also opposite to the original loop (Fig. S5e). The expectation and the correlations (Fig.   S5f) commonly arrive at .  figure. a b x 3 x 1 x 2 x 3 x 1 x 2 c d e f x 3 x 1 x 2 x 3 x 1 x 2 x 3 x 1 x 2 x 3

Section 7. Determination of the topological charge j without calibration
In the main text, the topological charge was derived by the calibration using the rotation matrix which decreases the angle between and -axis. Here, this charge is derived without the calibration. Because the closed loop in Fig. 3a in the main text contains both cyan and orange rings, a rotation matrix can be written as a composition form: gives the consistent topological charge as . . The red, green, and blue colors correspond to , 2, and . c, A closed loop with merged two nodal points. d, Plots of on the case of c. a b x 3 x 1 x 2 c d x 3 x 1 x 2 x 1 x 2 x 1 x 2

Wilczek-Zee connection
We start this discussion with the generalized Wilson operator. 1,3 If we consider a closed loop enclosing a nodal line and it is parametrized by ∈ 0, 2 , the generalized Wilson operator is written as 1,3 is the Berry-Wilczek-Zee (BWZ) connection. 4 The sub-and superscripts and are the band numbers ( , 3, 4, 5 for 3rd, 4th, and 5th bands, respectively) and is a periodic part of the magnetic field eigenstate • of the pth band such that .
From the results shown in Fig. 1b in the main text, we calculated and as plotted in Fig. S7a-b, respectively. From these results, the topological charges can be assigned on each nodal line; for the closed loop encircling the nodal line by the 3rd and 4th (4th and 5th) bands, only ( ) varies by while the other remains zero.