Sign freedom of non-abelian topological charges in phononic and photonic topological semimetals

The topological nature of nodal lines in three-band systems can be described by non-abelian topological charges called quaternion numbers. Due to the gauge freedom of the eigenstates, the sign of quaternion numbers can be flipped by performing a gauge transformation, i.e., choosing a different basis of eigenstates. However, the sign flipping has not been explicitly shown in realistic systems such as phononic and photonic topological semimetals. Here, we elaborate on the sign freedom of non-abelian topological charges by visualizing numerically calculated topological charges in phononic and photonic topological semimetals. For this, we employ a common reference point method for multiple nodal lines and thus confirm that the sign flipping does not cause any inconsistency in building the quaternion group.

Although the gauge dependence of the non-abelian charges has been already discussed in the literature [1,21,[43][44][45][46][47], such property has not been explained by visualizing the eigenstates. In any eigenvalue problem, if u is an eigenstate of a system, e iθ u (θ = (0, 2π ] ) is also an eigenstate of the system, and these two are regarded as the equivalent states (gauge freedom). Even if we restrict the gauge e iθ to be a real number, i.e., θ = 0 or π, the eigenstate of the system can be expressed either by u or −u. This means that the gauge freedom allows any signs of the topological charges in the nodal line system generated in the electronic (metals [1,10,15] and semimetals [11,12]), photonic [18,22], and phononic systems [7,9,48]. This property, however, does not contradict the argument of the non-abelian band topology in a three-band system, and it is important to fix the basis of eigenstates as pointed out in reference [1].
In this paper, we elucidate the gauge-dependent sign freedom of the non-abelian topological charges by visualizing the topological charges in phononic and photonic topological semimetals. First, we exemplify such a property with phononic and photonic systems. Then, the reference point method, which is equivalent to fixing of the basis in Hilbert space [1,45], is adopted for our topological semimetals. Finally, we discuss the consistency of the topological charges' non-abelian relations and the important points to be considered when using this method in phononic and photonic topological semimetals focusing on the path and degeneracies in momentum space.

Gauge-dependent property of topological charges
Obtaining the quaternion charge of a nodal line starts with calculating eigenstates along the closed loop that encloses the nodal line. We denote the starting point of the closed loop's winding as k 0 as shown in figure 1(a). Once the signs of the eigenstates at k 0 are determined, their signs at subsequent points k on the loop are assigned. Finally, the topological charges are calculated by analyzing the rotation behaviors of the eigenstates. Due to the sign freedom of the eigenstates at k 0 , the topological charge also has the sign freedom [1,21,[43][44][45][46][47]. In this section, we discuss the examples that show the sign freedom of the quaternion charges originating from the sign freedom of eigenstates at k 0 .

3 × 3 Hamiltonian
We consider phononic waves in an orthotropic elastic material [49] whose constitutive equation is given by Hooke's law ε = C · s (see equation (A.1)). Here, ε = ε ij and s = s ij (i, j = 1, 2, 3) are the Cauchy strain and stress tensors, respectively. C is a compliance matrix, which is a function of Young's moduli, shear moduli and Poisson's ratios: C = C E i , G ij , ν ij . Its inverse is the stiffness matrix S, i.e., C −1 = S, thereby the constitutive equation becomes s = S · ε which is analogous to the one-dimensional Hooke's law F = −kx. By using the density ρ, Young's moduli E i , shear moduli G ij , Poisson's ratios ν ij , and restoring forces f i (i, j = 1, 2, 3) listed in table A1 in appendix A, the eigenvalue problem for this elastic wave system is expressed as with where u n k (n = 1, 2, 3) is the three-component displacement eigenvector satisfying the orthonormal condition u m k · u n k = δ mn , and ω n is the angular frequency as an eigenfrequency (the detailed derivation of equation (1) can be found in appendix A). Here, the superscript n is the band number, and k is a point in the momentum space.
For the nodal link shown in figure 1(a) in this phononic system, we choose two closed loops (the arrowed loops) that enclose each nodal ring (the orange and cyan colored shapes, respectively). We calculate the orthonormal eigenvectors u n k along the closed loops [22]. We gather all the eigenstates at the origin, and their evolution in a three-dimensional space R 3 defined by the three components of u n k 's are shown in figures 1(b) and (c), respectively. We may regard the topological charge for figure 1(b) as the quaternion number k because u 1 k and u 2 k rotate by +π around fixed u 3 k [1,18,22,50]. Likewise, u 2 k and u 3 k rotate by +π around fixed u 1 k in figure 1(c), and thereby its topological charge is the quaternion number i [1,18,22,50].
Meanwhile, due to the sign freedom of the eigenstates, we can choose different eigenstates at k 0 . For the orange loop in figure 1(a), we flip the signs of u 1 k 0 and u 3 k 0 while keeping u 2 k 0 fixed. Then, u n k at the following points k on the loop is determined by u n k 0 . The resulting relation between the eigenstates u n k exhibits −π-rotations of u 1 k and u 2 k around u 3 k (see figure 1(d)). Thus, the topological charge for this situation is considered as −k, compared to figure 1(b). Likewise, we apply the sign-flipping of u 1 k 0 and u 2

Correlation vectors in photonic systems
Alternatively, the gauge-dependent non-abelian charges can be visualized using the correlation vectors [22]. For the closed loop in figure 2(a), we compute the orthonormal eigenstates |ψ n k (n = 1, 2, 3 for the third, fourth, and fifth bands, respectively). Then, we calculate the correlations C n k defined by the following equation [22]: All the correlations C n k on the closed loop are collected at the origin, as shown in figure 2(b). The result means its topological charge is −iσ 3 , equivalent to k. Now, we adjust the signs of ψ 1 k 0 and ψ 3 k 0 while fixing the sign of ψ 2 k 0 . The signs of |ψ n k at the remaining points k on the closed loop are assigned by the ψ n k 0 . The resulting topological charge in figure 2(c) is +iσ 3 , or −k.
Likewise, for the closed loop in figure 2(d), we consider two sets of ψ n k 0 . The sign of ψ n k 0 in the first set is not adjusted. For the second set, we flip the signs of ψ 1 k 0 and ψ 2 k 0 with fixing the gauge of ψ 3 k 0 . Thus, the topological charges of figures 2(e) and (f) are −iσ 1 and +iσ 1 , respectively, which correspond to i and −i, respectively.

Application of common reference point method
To unambiguously determine the signs of the topological charges of our topological semimetals, we fix the gauge of the eigenstates using a reference point [1,45]. Here, we denote the common reference point as k r , and a line connects k r and k 0 of each loop, as illustrated in figure 3. There can be several k 0 for several closed loops whereas we consider only one k r . When the topological charge on one closed loop in figure 3 is investigated, ψ n k r is calculated first. ψ n k r +Δk is also calculated at k r + Δk on the k r − k 0 line. The sign of ψ n k r +Δk is determined to make ψ n k r |ψ n k r +Δk positive. For all the remaining points k on the k r − k 0 line, the same calculations are performed; the sign of |ψ n k is determined by ψ n k r −Δk |ψ n k > 0. Once k reaches k 0 , ψ n k 0 is determined. Finally, the topological charge is calculated by the eigenstates only on the loop. Instead of using the reference point k r , one can choose the loops whose starting points k 0 coincide at one point in momentum space. However, a smaller loop generates a more accurate topological charge, due to the effect coming from the relative positions between the path and degeneracies that will be explained in section 5.1. Thus, this alternative approach can be employed in limited situations where the nodal lines are sufficiently close to each other.
For the phononic topological semimetals discussed in section 2.1, we set a common reference point k r = [−1, 1, 0.5] as shown in figure 4(a), and then we calculated the eigenstates of equation (1) figure 1(c)) resulting in topological charge +i. Here, the eigenstates at the common reference point k r for path A and B should be identical.
The topological charge j = ki can be described by the composition of the loops in path A and B, which are the closed loops encircling both orange and cyan nodal lines. Note that their starting points k 0 should  be connected to the common reference point k r . Here, the circling sequence is related to the order of k and i in the multiplication. If the first and second half of closed loop scan the cyan (i) and orange (k) nodal lines, respectively, the relation is written as ki. The signs of k and i are concerned with winding direction of the closed loop. As we can consider four different combinations in this multiplication, namely ki, ik, (−i) (−k), and (−k) (−i), their results are j, −j, −j, and j as shown in each panel of figure 5, respectively. The plots in figure 5 are the calibrated results following the step in reference [22]. The calibration angle was set as 0.8θ 0 (k) where θ 0 (k) is the angle between u 2 k and u 2 k 0 . Note that the topological charges k, i, and ±j of the double diamond photonic crystal in reference [22] were calculated using the reference point. More details will be explained in section 5.1.

Consistency of topological charges' signs by the reference point method
In the following, we will show that the reference point method can eliminate the chance that the non-abelian charges are misinterpreted as abelian. For example, let us suppose that two closed loops, which are marked as loop 1 and 2 in figure 6, respectively, are in different locations in momentum space. If we denote topological charges by these loops as N 1 and N 2 , respectively, and if we consider an additional loop, marked as loop 3a in figure 6, circling along loop 2 and 1 in sequence [18,22], the loop 3a's topological charge N 3a is calculated by the composition of N 1 and N 2 , i.e., N 3a = N 1 N 2 . We also suppose that loop 3b scans around loop 1 first then circles along loop 2, as shown in figure 6. Its topological charge is expressed as N 3b = N 2 N 1 . If N 3a and N 3b satisfy N 3a = −N 3b , this leads to N 1 N 2 = −N 2 N 1 . This relation can be called  'non-abelian'. To write this relation, the signs of each charge should be first determined by such kind of sign convention. If there is not any sign convention, and if the topological charge of loop 3b can be either plus or minus, it means that we can assume the sign-flipping of N 3b while fixing the other charges' signs. Then, the above relation N 3a = −N 3b becomes N 3a = N 3b or N 1 N 2 = N 2 N 1 . Thus, such a situation could be incorrectly interpreted as 'abelian'. This behavior is, however, simply a result coming from the gauge freedom of the eigenstates, and there is no inconsistency in the non-abelian nature of the non-abelian band topology.

k r − k 0 path and degeneracies
Although figure 3 illustrates the paths between k r and k 0 as a line, they do not have to be a straight line. Instead of the straight line (see figure 7(a)), the path sometimes should make a detour around degeneracies (see figure 7(b)) which can be zero-, one-, two-, or three-dimensional. Let us suppose we investigate the topological nature of three bands indexed as n, n + 1, and n + 2. Then, the paths should avoid the degeneracies that at least one of the three bands is concerned with, e.g., the degeneracies by the bands n − 1 and n or the degeneracies by the bands n + 2 and n + 3. This is because the degeneracies destroy the  [22]. The nodal link is formed by the third, fourth, and fifth bands (n = 1, 2, and 3, respectively), and the pink-colored degeneracies are formed by the second and third bands (n = 0 and 1, respectively). Two paths are also marked. Path 1 intersects the pink degeneracies near k 0 while path 2 does not. (d) and (e) ψ n k−Δk |ψ n k along paths 1 and 2, respectively.
eigenstates information at k r . In figure 7(c), path 1 goes through the pink points (near k 0 ) while path 2 avoids them. The pink points are degeneracies between the second and third bands (n = 0 and 1, respectively). ψ n k−Δk |ψ n k (n = 1, 2, 3) at each point on path 1 are calculated, and only the value with n = 1 shows the sharp change around the degeneracies (see figure 7(d)). In contrast, their behaviors on path 2 do not exhibit such a sharp change (see figure 7(e)). In other words, the eigenstates information at k r cannot be delivered through the degeneracies because the degeneracies may act as the sink or source of certain topological states, like Weyl points [33-42, 51, 52].
Importantly, the sign of the quaternion charges depends on which detour the path takes, as discussed in references [1,45]. In our elastic system, we considered k r located in front of the nodal link, as shown in figure 4(a). When the same reference point is used to figure 5, multiple detour options are available for setting k r − k 0 in figures 5(b) and (d), unlike the points k 0 in figures 5(a) and (c) that can be connected to k r along the shortest path. To calculate the topological charges in figures 5(b) and (d), we used straight k r − k 0 passing the center of the insets in figures 5(b) and (d). These paths go through both orange and cyan rings. If we choose a detour k r − k 0 that does not pass through any of these rings, the signs of the topological charges in figures 5(b) and (d) remain the same. If the path pierces through only one ring, the signs are flipped, which can be analyzed in depth following references [1,45].
Based on the above discussion, it is desirable to set the reference point k r that makes the overall k r − k 0 short as possible to reduce the chance of losing the Bloch states' gauge information. Especially, at the first step of calculating the quaternion charges, it is simple to set the k r − k 0 not to pass through nodal rings. If k r and k 0 are in opposite regions of degeneracies, the topological charge's signs should be examined using references [1,45].

Gauge freedom at the reference point
Although we have examined the topological charges of our topological semimetals using the reference point method, the exact sign on each topological charge is not fully determined due to the sign freedom of the eigenstates at k r . If the eigenstates' signs at k r are flipped, some quaternion charges' signs may be flipped while the other quaternion charges' signs do not change. For example, regarding figures 4 and 5, if we flip the signs of u 1 k r and u 2 k r at k r , the result in figure 4(e) may change from i to −i while the topological charge of figure 4(c) is still k. In this case, the anticommutative relations between the charges are still valid as the signs in figure 5 are also reversed.

Conclusions
In summary, we have shown that the sign of quaternion charges can be flipped due to the gauge freedom of eigenstates by using the phononic and photonic topological semimetals. We have also applied the reference point method, which is equivalent to the basis fixing, to our topological semimetals. Additionally, the consistency of non-abelian relationship between topological charges with this method has been discussed.
Based on the discussions in section 5.2, we make the following conclusions: (i) gauge fixing by a common reference point does not result in unique signs of the quaternion charges' signs but makes the non-abelian relations satisfied by the quaternion charges, (ii) the signs of nodal lines can be flipped by choosing the eigenstates' signs at a reference point k r . In other words, the sign relation between different nodal lines is more important than the absolute sign of quaternion charges corresponding to each loop.

Acknowledgments
The work is part-funded by the European Regional Development Fund through the Welsh Government (80762-CU145 (East)).

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.

Appendix A. Derivation of 3 × 3 Hamiltonian for an orthotropic elastic material
The constitutive equation for an orthotropic elastic material is expressed as ε = C · s, or where ε = ε ij and s = s ij are the Cauchy strain and stress tensors, respectively (i, j = 1, 2, 3). Each component of the Cauchy strain tensor is expressed as ε ij = 1/2 ∂u i /∂X j + ∂u j /∂X i where u = [u 1 , u 2 , u 3 ] is the displacement vector. C is the compliance tensor: where E i is the Young's modulus along i-direction and G ij = G ji is the shear modulus along i-direction on the plane normal to j-direction. ν ij is the Poisson's ratio, the negative ratio of a transverse strain along j-direction to a longitudinal strain along i-direction. Because we assume an orthotropic material, the relation ν ij /E i = ν ji /E j holds and the compliance tensor C is symmetric. The wave equation for an elastic material is −∇ · s − F v = −ρü, where F v is the body force. Here, we consider the body force as the restoring force expressed as F v = −f · u, i.e., F v = − f 1 u 1 , f 2 u 2 , f 3 u 3 . If the displacement vector is expressed as u (x, t) = u (x) exp (−iωt) = u k exp (−ik · x) exp (−iωt), substituting this into the wave equation leads equation (1). In equation (1), M ii and λ ij are the components of the  This orthotropic elastic material forms a nodal link when the parameters in table A1 are used. In the given the momentum space with −π k 1 π, −π k 2 π, and −π k 3 π, we calculated the eigenfrequencies ω n . We regarded that two bands n and n + 1 at point k are degenerate if ω n+1 − ω n a/ E 1 /ρ < 0.0268, where a = 1 m is the constant for normalization. Sets of the degeneracies formed between the first and second bands and the second and third bands are depicted as orange and cyan nodal lines, respectively, in figure A1(a). Band structures along k 1 and k 3 directions are also given in figures A1(b) and (c), respectively.