Topological near fields generated by topological structures

The central idea of metamaterials and metaoptics is that, besides their base materials, the geometry of structures offers a broad extra dimension to explore for exotic functionalities. Here, we discover that the topology of structures fundamentally dictates the topological properties of optical fields and offers a new dimension to exploit for optical functionalities that are irrelevant to specific material constituents or structural geometries. We find that the nontrivial topology of metal structures ensures the birth of polarization singularities (PSs) in the near field with rich morphologies and intriguing spatial evolutions including merging, bifurcation, and topological transition. By mapping the PSs to non-Hermitian exceptional points and using homotopy theory, we extract the core invariant that governs the topological classification of the PSs and the conservation law that regulates their spatial evolutions. The results bridge singular optics, topological photonics, and non-Hermitian physics, with potential applications in chiral sensing, chiral quantum optics, and beyond photonics in other wave systems.

The concept of topology has provided new perspectives for physicists to explore unconventional properties of physical systems, such as the one-way edge states in topological insulators and their counterparts in classical wave systems that have attracted considerable interest recently 1,2 .These properties are associated with the topology of the momentum space (i.e., Brillouin zone).Topology in the real space can also give rise to intriguing physical properties and phenomena.In particular, Maxwell's equations allow unique solutions with nontrivial topology in the real space, where the field lines, phase singularities (i.e., optical vortices), or polarization disclinations can form links and knots [3][4][5][6][7][8][9][10][11][12] .These topological configurations of electromagnetic fields can enable highly flexible manipulations of phase and polarization with unprecedented precision for various applications.
At an arbitrary point of the three-dimensional (3D) real space, the end of electric/magnetic field vector of a generic monochromatic electromagnetic wave traces out an ellipse, i.e., the field is elliptically polarized 13 .The distribution of the polarization ellipses can form topological defects known as polarization singularities (PSs) 14 , which include C points (where the field is circularly polarized and the direction of the major axis of polarization ellipse is ill-defined), L points (where the field is linearly polarized and the normal direction of polarization ellipse is ill-defined), as well as V points (where the field norm is zero and the field direction is illdefined).The 3D lines formed of the PSs are referred to as C lines, L lines, and V lines accordingly.These polarization singularity lines (PSLs) can emerge during light focusing 9,15 , scattering [16][17][18][19][20] , interference 10,21 , in nanostructures including metasurfaces 22 and photonic crystals 23 .The integration of singularity and topology theory has revealed richer physics including geometric phases 24,25 , bound states in the continuum (BIC) [26][27][28] , Hermitian topological nodal degeneracies 29,30 , and non-Hermitian exceptional points (EPs) [31][32][33][34][35] , etc.
The explicit geometry of optical structures decides the local resonance of optical modes and gives rise to novel optical devices and wave-functional materials, such as nanoantennas 36 , metamaterials 37 , and metasurfaces 38 .As a result, conventional studies in photonics mainly focus on the geometry and rarely pay attention to the overall topology of the structures investigated.It thus becomes interesting to ask: What optical properties are determined solely by the overall topology of optical structures?How can the topology of optical polarization fields and the topology of optical structures be interconnected?
In this article, we establish a universal and exact connection between the topology of optical fields and that of optical structures, revealing how the births and topological evolutions of magnetic PSs in the near fields are bounded by the topology and symmetry of the structures.
As is required by the Poincaré-Hopf theorem, the existence of the PSs is topologically protected, and they are characterized by quantized topological indices with the index sum solely decided by the genuses of the structures.We further demonstrate that, by incorporating extra spatial symmetries (such as the mirror symmetry and the generalized rotational symmetry) for the structures and the incident fields, higher-order PSs/PSLs and topologically stable nexuses of PSLs can emerge, around which the polarizations evolve into striking configurations such as mirror-symmetric double-twist Möbius strips [39][40][41] .To grasp the underlying invariant properties of the continuous evolutions of PSs (e.g., merging, bifurcation, and topological transition), we map the real-space C points to the EPs of a 2-band non-Hermitian Hamiltonian, upon which homotopy theory can be directly employed to identify the core invariant to classify all topological evolutions.

Results
Polarization singularities protected by surface topology of structures A general monochromatic magnetic field can be expressed as  = ( + ) , where  and  are the major and minor axes of the polarization ellipse, and  = (𝐇⋅𝐇) is a proper phase.The C points of the magnetic field  correspond to the phase singularities of the scalar field Ψ =  •  and generally can form stable C lines in 3D space without any symmetry protection.A C line can be characterized by two topological indices (i.e.winding numbers): the polarization index  = ∮ and the phase index  = ∮ ∇ ⋅ , where  is the azimuthal angle on the Poincaré sphere for the in-plane polarization (i.e., in the plane of the polarization ellipses at a C point) and  = Arg(Ψ), and both integrals are evaluated on a small loop enclosing the C line 25,42 .We note that  is uniquely defined at each C point provided that the magnetic spin  = Im( * × ) is not normal to the C line.At the points where  is normal to the C line,  may change sign, making  not a global invariant along the C line.In contrast, although the sign of  depends on the direction of the integration loop, it is invariant against continuously moving the loop along the C line.
Therefore,  can endow the C line with a positive direction  = sign  , where  is the tangent vector of the C line complying with the right-hand rule of the integration loop of  43 .One can prove that the two indices are related by  = sign( ⋅ ) /2 (see Supplementary Note 2).
Let us first consider a metal sphere under the incidence of a plane wave propagating in z direction and the magnetic field is linearly polarized in y direction.Without loss of generality, we assume that the metal is gold characterized by a Drude model (see Methods).We conducted full-wave numerical simulations of the system by using a finite-element package COMSOL Multiphysics.The numerically obtained PSLs at the frequency  = 100 THz are shown in ).Although a V line cannot stably exist in general, the intersections of two C lines can form stable V points protected by mirror symmetry, as will be proved later.
We now ask the question: what determines the morphologies of the PSLs and their topological indices?It turns out that the answer is rooted in the topological properties associated with the geometry of the metal spheres.Under the excitation of the incident electromagnetic field, currents are induced in metal structures.These currents mainly localize within a thin surface layer of the structures, the thickness of which is approximately equal to the "skin depth" (i.e., the depth that light penetrates the metal) 44 .By the boundary conditions, the magnetic field H near the metal surface is dominated by its tangent component (see Supplementary Note 1).Consequently, the major axis A of the polarization ellipses can be considered a tangent bi-vector field (i.e., line field) defined on a two-dimensional (2D) smooth manifold M (i.e., the surfaces of the structures).According to the Poincaré-Hopf theorem for tangent line fields, a finite number of isolated singularities of this field must emerge on the surfaces of the structures 45 .These singularities are exactly the C points and/or V points, and the sum of their topological polarization indices must satisfy ∑  ( )  = (), where  denote the PSs on the smooth manifold M, and  denotes the Euler characteristic of M. For a closed orientable surface,  is directly given by the genus g of the surface:  = 2 − 2 45 .For the single sphere case in Fig. 1 again agreeing with the Euler characteristic of the double-torus.We have also simulated the case of triple-torus ( = 3) and the results also satisfy the theorem (data not shown).These results, for the first time, establish a direct relation between the topology of optical structures and the topological properties of optical near fields.This relation holds for arbitrary metal structures as long as their geometric surfaces are smooth and the skin depth is small.
Since the emergence of PSLs is protected by the topology of the structures, their global topological properties are robust against continuously varying the geometry of structures unless singular perturbations are introduced to the geometry.For example, if we introduce sharp edges into the sphere such that the radii of curvature near the edges are comparable with the skin depth, the total index of PSLs can change, as shown in Fig. 2(a), where ∑  = 0.The presence of sharp edges (where the tangent planes and thus the tangent fields are not well defined) renders the original Poincaré-Hopf theorem for smooth manifolds inapplicable to the surface.Thus, the total index is not necessarily determined by the genus of the geometry.On the other hand, if the sharp edge is "smoothed out", as shown in Fig. 2(b), the global topological properties of the PSLs recover, i.e., the indices satisfy the theorem again.A second example is given in Fig. 2(c), where a cylindrical portion is removed from the sphere.In this case, the sharp edge induces two additional C lines with  = − , and the total index of the C lines is zero.After smoothing the edges, the total index recovers the value of +2, as shown in Fig.

Mapping to non-Hermitian exceptional points
The PSLs evolve as they extend away from the surfaces of the structures, leading to merging, bifurcation, and topological transition in the 3D space.To understand these phenomena, we employ a mapping from the C points (real-space singularities) to the non-Hermitian EPs (parameter-space singularities).We introduce an auxiliary 2-band non-Hermitian Hamiltonian associated with the magnetic field, ℋ() = () •  ⃗, where  ⃗ =  ,  ,  is the vector of the eccentricity of the polarization ellipse 1 − and are degenerate when the discriminant of the Hamiltonian's characteristic polynomial reduces to zero:  = (ℎ − ℎ ) = 4 •  = 4Ψ = 0, i.e., the condition for the emergence of a C point or a V point.Thus, the degeneracies of the non-Hermitian Hamiltonian, i.e., EPs and non-defective nodal points, just correspond to the C and V points of the magnetic field, respectively.This remarkable property allows a mapping from the topology of EPs given by the 2-band non-Hermitian Hamiltonian to the topology of C points in magnetic field.By borrowing the topological classification of the 2band non-Hermitian Hamiltonian with separable bands 46 , we can obtain the topological classification of the PSLs in the 3D real space.Specifically, in the absence of any symmetry constraint, the configuration space of the non-Hermitian Hamiltonian without degeneracies can be expressed by the coset space 46  ≃ ( ×  )/ℤ (see Methods), which can be physically understood as the configuration space of the magnetic field in the real space:  stands for the orientation sphere of the major axis A of the polarization ellipse;  corresponds to the circle of the phase  = arg(Ψ); ℤ denotes the redundancy that both the direction of A and  change sign simultaneously.We note that since only C and V points are of our interest, the vanishing of the minor axis B of the polarization ellipse (i.e., the condition of L points) is irrelevant to the topological classification here.The first homotopy group of the configuration space X gives the topological classification of the polarization field along any closed loop in the space 46  the phase winding number of Ψ along a loop , i.e., phase index  ().It corresponds to the energy vorticity, also known as the discriminant number, in the notation of non-Hermitian physics [33][34][35] .Therefore, the phase index  () is not only conserved against continuous deformation of the loop, but also a complete index that can characterize all topological phases associated with C lines.In comparison, the polarization index defined along an arbitrary loop, characterized by the trivial or Möbius twists of the major axis A, is not a complete index (see Supplementary Note 2).As we have shown previously, if a loop only encloses one C line, the phase index  defined with the loop can assign a positive direction  to the C line.More broadly, the topological index  () defined with an arbitrary loop  counts the net number of directed C lines passing through the loop.
Since the topological invariant ℤ is equal to the phase index  , we can apply this invariant index to understand the topological transition of PSLs.Consider the double-sphere case in Fig. 1(b), at large separation of the spheres, the PSLs must reduce to that of two isolated spheres (corresponding to two copies of Fig. 1(a)).This involves a topological transition.

Integration of topology and mirror symmetry
To further understand the properties of the PSLs in Fig. 1, it is necessary to discuss the combined effect of mirror symmetry and topology.If we impose a y-mirror symmetry about the mirror plane y = 0, marked as Π, it is straightforward to show that the 2-band Hamiltonian satisfies ℋ() = − ( ) ⋅  ⃗ =  ℋ( ) , where  = diag(1, −1,1) denotes the mirror reflection operator for polar vectors about y = 0. Thus, the magnetic field in a mirrorsymmetric system can be mapped to a Hamiltonian with a mirror symmetry.In this case, the magnetic field on the mirror plane Π only has perpendicular component, (, 0, ) =  (, 0, ) , hence the configuration space  of the nonsingular magnetic field on Π is given by  =  =   |  ≠ 0 ≃ ℂ − 0 ≃  .Therefore, we obtain the topological classification of the magnetic fields along the loops in Π according to the first homotopy group:  ( ) =  ( ) = ℤ, which is characterized by the different phase winding number of  along the loops.In this case, topologically stable V points just appear at the phase singularities of  in the mirror plane.The phase index of each V point defined on a loop encircling the V point in the mirror plane must be  = ∮ ∇ Arg  ⋅  = ∮ ∇ Arg  ⋅  = ±2.The conservation of the phase index  = ±2 indicates that such a mirror-symmetry-protected V point is not an isolated singularity but manifests as the intersection point of two C lines that pierce the mirror plane from the same side (see Fig. 4).
We now apply the above results to understand the V points labeled as V1 and V2 in Fig. Möbius strip protected by mirror symmetry.We note that the absolute times of winding of 3D polarizations are not a topological invariant, which can change arbitrary even number of times by deforming either the loop or the system without breaking the symmetry 25,39 .Thus, here the number of twists should be counted in a topologically stable sense (i.e., the minimal number of twists that cannot be untied by continuous deformation).In addition, a mirror-symmetryprotected double-twist (trivial) polarization strip must enclose odd (even) number of C lines at each side of the mirror plane (see Supplementary Note 3).The above analysis is confirmed by the numerical results in Fig. 4, where the pink arrows denote the polarization major axes A on a TSMS loop for the V1 and V2 cases.We notice the mirror-symmetric double-twist Möbius strips in both cases.This also explains why the two oppositely directed C lines at the two sides of the mirror plane cannot annihilate but must form a V point when meeting on the mirror plane.
Once the mirror symmetry is broken, the double-twist Möbius strip can be deformed into a trivial strip with no twist, and the two C lines will be gapped at the V point.Therefore, our study reveals that spatial symmetry can engender intriguing topologically nontrivial polarization configurations that cannot stably exist in the non-symmetric case.

Integration of topology and generalized rotational symmetry
Higher-order PSs/PSLs are generally unstable without symmetry protection and can easily transform into multiple lowest-order PSs/PSLs under perturbations 50 .The V points in Fig. 4 are a type of higher-order PSs protected by mirror symmetry.It is interesting to explore whether higher-order C points/lines can appear if the topology of structures is integrated with rotational symmetry.
We consider the metal sphere under the excitation of a right-handed circularly polarized For  = 1,2, the minimal charge is  = 0, which is consistent with our prediction that there is no C line on the y axis; for  = 3, the central C line is first order with  = = −1/2; for  = 4, the central C line is second order with either  = +1 or  = −1; for  ≥ 5 , the C line must be second order with positive index  = +1 (see Supplementary Note 4 for an alternative proof via perturbation analysis).Akin to the cylindrical symmetry case, a left-handed circularly polarized incident wave illuminating a  symmetric scatter will induce opposite  and  but an identical  , in comparision to those generated by a righthanded circularly polarized incident wave.This can also be understood from the compatibleness between the line field configurations and the rotational symmetries (see Supplementary Note 4).
As the most interesting case, we generate the central C lines with  = ±1 protected by the  ̅ rotational symmetry, using the torus nexus with genus  = 3 in Fig. 5(c).Under the incidence of the same circularly polarized plane wave, this structure generates complexly distributed C lines.For the ease of visibility, we show the C lines without the torus nexus in C lines is also confirmed via the perturbation analysis near the  axis (see Supplementary Note 4).Still by the conservation of  and  ̅ symmetry, the four "budding" C lines are first order and must be directed either all outward or all inward such that the total "arrows" weighted by  flowing to each nexus point are equal to zero, as shown by the smaller yellow arrows in the inset of Fig. 5(d).

Discussion
PSs can emerge in various parameter spaces.In momentum space, they are determined by the Bloch states and are intimately related to the BICs in periodic photonic structures, where V points can give rise to vanishing far-field radiation of Bloch states 53,54 .PSs also provide a direct way to visualize the band Chern number and the topological structure of EPs 32,55 .Such studies typically focus on the PSs that live on the Bloch torus associated with a 2D periodic structure.
Extension to other parameters spaces of different topology (genus) is difficult, if not impossible.
In the real space, the PSs are attributed to the interference of the Fourier plane waves of the incident and scattering fields.This endows the real-space PSs with several unique properties.
They can live on various 2D manifolds and give rise to intriguing PSLs whose configurations can be easily controlled via the topology and symmetry of optical structures, as we have demonstrated here.In addition, the PSs in 3D space naturally carry Pancharatnam-Berry phase and spin-redirection phase 25,56 , offering rich mechanisms for manipulating light's polarization and phase.Because of their intrinsic chirality deriving from spin and inhomogeneous phase distribution in the near field, the emergence of the PSs is usually accompanied by the spin-orbit interactions of light 57 .Such spin-orbit interactions can enable directional near-field coupling and far-field radiation with fruitful applications in novel optical sources and nanoantennas 58,59 .
Experimentally probing these PSLs can be achieved by using the near-field characterization techniques that can map subwavelength longitudinal and transverse field components, such as scanning nearfield optical microscopy 60 .
In sum, we establish a direct relationship between the topology of metal structures and the magnetic PSLs in the near field.We show that the index sum of the PSLs born on the surface of the structures is solely determined by the Euler characteristic of the structures due to the tangent nature of magnetic field near the metal surface.In addition, we find that the interplay of topology with mirror symmetry or generalized rotational symmetry cam give rise to topologically stable higher-order PSs/PSLs lines with richer morphologies including C line nexuses and mirror-symmetric double-twist Möbius strip of polarizations.Remarkably, the topological properties of the PSLs can be well understood by using a non-Hermitian 2-band Hamiltonian, where the topological classification of non-Hermitian EP lines can be mapped to that of C lines characterized by the phase index.According to this correspondence, the merging, bifurcation, and topological transition of C lines in the real space have to observe the phase index conservation law.Our study uncovers exotic topological properties of optical polarization fields that are irrelevant to the specific material or geometry of the optical structures.The results have essentially connected polarization singularities, the topology and symmetry of structures, as well as non-Hermitian physics, opening extra opportunities for fundamental conceptual explorations and many related practical applications in chiral discrimination and sensing, chiral quantum optics, and topological photonics.They may also be extended to other classical wave systems such as sound waves and water-surface waves 61,62 .

Numerical simulations
Full-wave numerical simulations were performed by using a finite-element package COMSOL Multiphysics 63 .The considered structures are made of gold characterized by a Drude model , where  = 1.28 × 10 rad/s and  = 7.10 × 10 rad/s 64 .
The spheres have radii of 500 nm.The inner and outer radii of the single torus in Fig. 1(c) is  = 500 nm and  = 1500 nm.For the double-torus in Fig. 1 (d), we set  = 250 nm and  = 750 nm, and the center distance between the holes is 500 nm.In all cases of Fig. 1, we assume a plane wave propagating in z direction, and the magnetic field is linearly polarized in y direction, and open boundary conditions are applied.In the cases of Fig. 5, we assume that the incident plane wave propagates in y direction and is circularly polarized.

Homotopy approach characterizing the topology of C lines
According to the approach for classifying the gapless topological phases of non-Hermitian Hamiltonians 46 , the degenerate submanifold  (i.e.EPs and non-defective degeneracies) is regarded as the topological obstruction in the configuration space of the Hamiltonians  ℋ = {ℋ =  •  ⃗:  ∈ ℂ } , the existence of which induces the nontrivial connectivity to the subspace of  ℋ excluding the degeneracies  =  ℋ −  = {ℋ ∈  ℋ :  ⋅  ≠ 0 } (i.e. the subspace with separable bands).Tracing a closed loop in , if the loop is noncontractible via continuous deformation, it must encircle the degenerate submanifold in an inextricable manner.
Therefore, the topology of the EP lines is essentially encoded by the topological classification of the loops in the nondegenerate subspace , mathematically given by the first homotopy group of .Homotopy refers to a topological equivalence relation.For two loops in  that can be continuously transformed to each other, they belong to the same homotopy equivalence class and hence characterize the same topological phases of degeneracies.And the first homotopy group is just the group formed by all homotopy equivalence classes of loops in X.
The configuration space of the non-Hermitian Hamiltonian without degeneracies is given by 46 , where the first part formed by the coset space of the complex general linear groups  (ℂ) represents the space spanned by the two eigenstates of the Hamiltonian, the second part Conf (ℂ) = {ℎ , ℎ } ∈ ℂ: ℎ ≠ ℎ denotes the configuration space of two separable complex eigenvalues of ℋ(), and the divisor ℤ denotes the redundancy that the eigenstates and eigenvalues change the orders simultaneously.
Moreover, it is shown that the eigenstate and the eigenvalue parts are homotopy equivalent to a sphere and a circle, i.e.
(ℂ) (ℂ)× (ℂ) ≃  and Conf (ℂ) ≃  , respectively 46 .Therefore, the configuration space for topological classification can be rewritten as  ≃ ( ×  )/ℤ      compute the relative homotopy group  (,  ), we consider the natural fibration of the configuration space , which is defined by the projection map from the configuration space  to the space of the major axis ℝ [6,11]: with  () = 0 or 1 ∈  (,  ) = ℤ .Comparing with Eq. (S4), we know that if the ymirror symmetry is broken, both the trivial and nontrivial phases in Eq. (S14) will reduce to the trivial phase.Therefore, the mirror symmetry is crucial to preventing the double-twist strip from unknotting.

Fig. 1 (
Fig.1(a).Interestingly, a pair of C lines emerge near the surface of the sphere.For each C line, (a), there are four C points on the sphere surface with the same index  = + , thus, ∑  = + × 4 = +2 = , consistent with the Poincaré-Hopf theorem.For the case of coupled spheres in Fig.1(b), there are four C points and two V points on the spheres' surface with total index ∑  = + × 4 + 1 × 2 = +4 =  , again satisfying the theorem.To further verify the above interpretations based on Poincaré-Hopf theorem, we calculated the PSLs generated by metal structures with genus  = 1 and  = 2 as shown in Fig.1(c) and (d), respectively.The same incident plane wave is applied in Fig.1(a)-(d), and we focus on the frequency range with small skin depth.Under this condition, the physics is general and is not restricted to a particular frequency.For the single torus in Fig.1(c), in addition to two C lines, we observe an accidental V line with  = −1 (green-colored) emerged inside the hole, corresponding to a pair of degenerate C lines with  = − .The index sum of the PSLs for the single torus is ∑  = + × 4 + (−1) × 2 = 0, which is equal to the Euler characteristic of a torus.Figure1(d)shows the PSLs generated by the double-torus with genus  = 2.Remarkably, a rich structure of the PSLs appear with crossings of C lines.There are totally twenty C points on the metal surface with total indices ∑  = + × 8 + − × 12 = −2, which essentially classifies the topologically different morphisms of the PSLs (i.e., C lines in general) encircled by the loop.The ℤ topological invariant distinguishing different homotopy equivalence classes in  () is just

Figure 3 (
Figure 3(a)-(d) shows the PSLs for the double-sphere case when the separation is increased,

3
(c), (d), (g) and (h).Additionally, the results in Fig.3indicate that C lines with opposite polarization index  do not necessarily annihilate because their chirality (i.e., spin S) can be different.An example is the C lines 1 and 4 in Fig.3(b) and (f), which have opposite  and opposite chirality.On the other hand, this can be easily understood based on the phase index since both C lines carry  = −1.
1(d), which are nexuses of two mirror-partner C lines protected by a y-mirror symmetry of the system.Figure4(a) shows the polarization ellipses and the phase Arg(Ψ) near V1 on a cutting plane and the y-mirror plane, respectively.The polarization ellipses indicate that the two C lines have opposite chirality ( ) = − () (denoted by the blue and red colors of the ellipses) but same polarization index  = +1/2 (denoted by the red color of the C lines), as guaranteed by the mirror symmetry (see Supplementary Note 3).As a result, the two C lines are oppositely oriented about the mirror plane, i.e.,  ( ) = − (  ) according to the relation  = sign   ⋅  , as shown by the yellow arrows on the C lines.In other words, both  and  behave as pseudo-vectors under mirror reflection.The phase Arg(Ψ) on the y-mirror plane has a −4 variation around the V1 point, corresponding to a phase index  = −2 (viewed from + direction) and in accordance with the fact that the two C lines both point inward (− direction) from the mirror plane at the V1 point.Figure 4(b) shows the polarization ellipses and the phase Arg(Ψ) near the point V2 on a cutting plane and the ymirror plane, respectively.Similar to the case of Fig. 4(a), the two C lines have opposite chirality but the same polarization index  = −1/2 and both puncture inwardly through the mirror plane at the V2 point, which is consistent with the phase index  = −2 around the V2 point as shown by Arg(Ψ) on the mirror plane.It is known that interesting Möbius strips of polarizations appear around a single C line8,9,[39][40][41][47][48][49] . Generaly, a polarization strip with odd number of twists must be topologically nontrivial, while a stripe with even number of twists can always be deformed to a cylinder with trivially aligned polarizations[25].An interesting question is: Could mirror symmetry enrich the topological classification and give rise to nontrivial polarization structures around the nexus of C lines?To explore this question, let us consider the polarization major axes A on a transverse self-mirror symmetric (TSMS) loop (i.e., a loop that intersects with and is symmetric about the y-mirror plane).Since the two mirror-partner C lines have opposite directions, the phase index for this loop vanishes, which seems to indicate a trivial topology on the loop with boring spatial structure of A. However, under the constraint of mirror symmetry, the topological classification along such TSMS loops is determined by the relative homotopy group  (,  ) =  × ℤ ,  = ℤ (see Supplementary Note 3).Consequently, despite carrying zero phase index, the TSMS loops can still have a nontrivial ℤ topology manifesting as a mirror-symmetric double-twist Möbius strip of the polarization major axes .Concretely, the major axis  on the mirror plane must point in the ± direction due to the y-mirror symmetry.The evolution of A on the TSMS loops gives rise to only two possible cases: trivial and nontrivial topological loops.Along a trivial loop, the polarizations can be continuously deformed into an untwisted strip.In contrast, the polarizations along a nontrivial loop must be twisted once at each side of the mirror plane and hence form a mirror-symmetric double-twist Fig. 5(b).However, the vanished Euler characteristic ( = 0) of the torus makes the C line

Fig. 5 (
Fig. 5(d).A total of twenty C points appear on the surface of the torus nexus: 4 points with

Fig. 2
Fig. 2 Effect of singular edges.C lines generated by a sphere with singular edges ((a), (c))

Fig. 4 V
Fig. 4 V points due to mirror symmetry.(a) V1 point due to the crossing of two C lines with

Fig. 5
Fig. 5 Higher-order C lines due to the generalized rotational symmetry.Higher-order C

Fig. S2 .Since
Fig. S2.Schematic of the relations between local polarization index, global polarization index, and phase index around a C line (blue).(a) Phase vortex and polarization ellipses lying on the plane perpendicular to the magnetic spin  (red arrow) of the central C point on a C line.(b) The Möbius strip of polarization ellipses along a finite loop  (white circle) around a C line.
S13)such that () =[±] gives the bivector of the major axis of the polarization ellipse for each point  ∈  , and the fiber at each point [±] ∈ ℝ :  ([±]) =  =   ∈ [0,2] ≃  ≃ (1) .Therefore, the projection :  → ℝ defines a (1)-principle bundle on the base space ℝ (i.e. the "polarization sphere" of the major axis, as shown in Fig.S3(d)where any two antipodal points on the sphere are identified as the same point in ℝ ).According to Theorem 4.41 in Ref.[6] for a fiber bundle :  →  that the  −homotopy group of the bundle  relative to the fiber  is isomorphic to the  −homotopy group of the base manifold by the induced map  * :  (, ,  ) →  (,  ) with  ∈  ( ), we obtain the topological classification on the semi-loops protected by -mirror symmetry ,  =   ×  ℤ ,  = *  (ℝ ) = ℤ .(S14)Recalling that ℝ is just the space of the major axis, Eq. (S14) reveals that the topology along semi-loops terminated in the mirror plane Π is absolutely determined by the twist of the major axes along the semi-loops, which are classified into two phases characterized by the two kinds of homotopic inequivalent trajectories on the ℝ sphere.Since the major axes at the two terminal points of a semi-loop is fixed along the normal direction of Π corresponding to the south and north poles of the ℝ sphere, the trajectories of  on the ℝ sphere should either form arcs connecting the two poles (these arcs is already closed in ℝ ), or form loops that start and end at the same pole.For the arcs on the ℝ sphere, the major axis exhibits an inextricable odd twist along the semi-loops (see Fig. S3(b)), thereby deemed as topologically nontrivial phase.In contrast, since the loops on the ℝ sphere are contractible to a point, the major axes A along the semi-loop exhibit a trivial twist (see Fig. S3(c)) and can be continuously deformed into the configuration that all A are parallelly aligned.Equivalently, along a selfmirror-symmetric loop  (the corresponding semi-loop is ), the major axes in the trivial and nontrivial phases form an untwisted strip and a mirror-symmetric double-twisted Möbius strip, respectively, which can be characterized by the global polarization index  () = 2 () = (0 or 2) mod 4, (S15)

2 .
Relation between mirror-symmetry protected ℤ  topology and C lines Now we discuss the relation of the twist of the polarization vectors along the semi-loops, and the number of the C lines at one side of the mirror plane enclosed by the semi-loops.First, we connect the two end points  ,  of the semi-loop  with an arbitrary curve  ⊂ Π, and hence obtain a closed loop  =  ∘  (see Fig. S3(a)).Severing the loop  at the base point  , we lift the major-axis bivectors and phase ,  ∼  with fixing a definite direction of  and a definite value of  at each point along the severed path  such that they are continuous along the path except for the breakpoint  (see Fig. S3(b)).Along  , ( ∈ ) remains fixed to  or − , i.e., we have (∈ ) |(∈ )| ≡  | | with  = ( ) .On the other hand,  is globally single-valued, ( ) ( ) =   =   where  ,  and  ,  denote the lift vectors and phases at the starting and end points of  (see Fig. S3(b)).These two facts lead to the equality exp   = exp[( −  )] = sign( ⋅  ) = sign( ⋅  ) = exp  () (S16) and equivalently,  () =   mod 2. (S17) This result reveals that (1) in the trivial twist phase of the major axis with  () = 0, the number of C lines on one side of  enclosed by the semi-loop  is even; (2) in the Möbius twist phase of the major axis with  () = 1, the number of C lines on one side of  enclosed by  is odd.This relation is illustrated in Fig. S3(b) and (c) for nontrivial and trivial semiloops, respectively.We also note that the arc connecting the two terminal points can be arbitrarily selected.For two selections  and  (see Fig. S3(a)), the corresponding closed loops are  =  ∘  and  =  ∘  .Their phase indices can be different, providing that the closed path  ∘  encircles some V points in the plane Π:   =   +  ( ∘  ).(S18) However, since  ∘  ⊂ Π must carry an even phase index, the parity of the phase index along  ∘  for arbitrary  are always invariant   =   mod 2, (S19)which is consistent with Eq. (S17) and confirms that the topological classification along the semi-loops is well-defined.

Fig. S4 .
Fig. S4.Streamlines of major axis in the transverse plane near the stable central C points protected by  ̅ symmetries.