Abstract
Oneway waveguides have been discovered as topological edge states in twodimensional (2D) photonic crystals. Here, we design oneway fiber modes in a 3D magnetic Weyl photonic crystal realizable at microwave frequencies. We first obtain a 3D Chern crystal with a nonzero first Chern number by annihilating the Weyl points through supercell modulation. When the modulation becomes helixes, oneway modes develop along the winding axis, with the number of modes determined by the spatial frequency of the helix. These singlepolarization singlemode and multimode oneway fibers, having nearly identical group and phase velocities, are topologicallyprotected by the second Chern number in the 4D parameter space of the 3D wavevectors plus the winding angle of the helix. This work suggests a unique way to utilize highdimensional topological physics using topological defects.
Introduction
Topological photonics^{1,2,3,4} started with the realization of oneway edge waveguides^{5,6,7,8,9} as the analog of chiral edge states of the twodimensional (2D) Chern insulator (or the 2D quantum Hall effect [QHE]), where the number and direction of 1D edge modes are determined by the 2D bulk topological invariant: the first Chern number (C_{1}). Threedimensional (3D) bands of nonzero C_{1} have also been realized in Weyl photonic crystals^{10}, opening doors to 3D topological phases for photons^{11,12}.
Here, we show that, by annihilating a single pair of Weyl points with helix modulations, light can be guided unidirectionally in the core of 3D photonic crystal fibers (Fig. 1), where the number and direction of oneway modes equal the magnitude and sign of the second Chern number (C_{2})—the topological invariant of complex vector bundles on 4D manifolds. This novel approach to create the linedefect states in the 3D topological bandgap provides a definitive way to obtain arbitrary mode number (C_{2} = −∞ to +∞) in the oneway fibers by varying the helix frequencies. Furthermore, all the modal dispersions have almost identical group and phase velocities, superior for multimode operations. The same phenomena can be realized in other Weyl systems^{13,14,15,16,17} with timereversal symmetry breaking.
Results
Single Weyl dipole
Our starting point is a photonic crystal containing two Weyl points^{18}, which were found in the double gyroid (DG) made of magnetic materials. The DG is a minimal surface that can be approximated by the isosurface of a single triply periodic function: f(x, y, z) = sin(2πx/a)sin(4πy/a)cos(2πz/a) + sin(2πy/a)sin(4πz/a)cos(2πx/a) + sin(2πz/a)sin(4πx/a)cos(2πy/a). This definition, although having a different form, yields almost identical geometry and band structure to those of the DG defined in ref. ^{18} by two separate trigonometric functions (one for each gyroid). In Fig. 2a, two cubic cells of the DG are shown, where f(x, y, z) > f_{0} = 0.4 is filled with gyroelectric material of dielectric constant \(\epsilon = \left( {\begin{array}{*{20}{c}} {17} & {  6i} & 0 \\ { + 6i} & {17} & 0 \\ 0 & 0 & {16} \end{array}} \right)\) and unity magnetic permeability. The rest of the volume is air. In this structure, there exists only two Weyl points (a single “Weyl dipole”) separated by about half of the Brillouin zone along z direction, as plotted in Fig. 2b. This means that an infinitesimal supercell modulation of the crystal (in z with a period of 2a) can superimpose the two Weyl points on top of each other to form a 3D Dirac point between four bands^{11} (Fig. 2b), which opens a gap under a finite modulation/coupling strength (Fig. 2d). The fact that a bandgap does not close under small perturbations ensures the robustness of this approach: certain mismatch between the Weylpoint separation and the wavevector of the modulation can be tolerated.
3D Chern crystal
We create a doublecell periodic modulation along z to annihilate the Weyl points and obtain the 3D Chern crystal, the photonic analog of the 3D Chern insulator (or the 3D QHE)^{19,20,21}. (So far, experimental realization of 3D Chern insulators is only limited to quasi2D systems^{22}). This modulation can be implemented in various system parameters, such as volume fraction, refractive index, magnetization, or structural distortion. In this work, we modulate the volume fraction of the DG by modifying the DG equations as follows: f(x, y, z) > f_{0} + Δf cos(πz/a), in which Δf = 0.07. The modulated DG is shown in Fig. 2c and its band structure is plotted in Fig. 2d.
A 3D Chern crystal is characterized by three first Chern numbers \({\bf{C}}_{\bf{1}} = \left( {C_1^x,C_1^y,C_1^z} \right)\) defined on the \(\hat x\), \(\hat y\), and \(\hat z\) momentum planes. For example, \(C_1^z\) is defined as
Because the bulk spectrum is gapped, \(C_1^z\) cannot change as a function of k_{z}. When there are N bulk bands below the bandgap, \({\cal F}_{xy}\) is an N × N matrix, whose elements are \({\cal F}_{xy}^{\alpha \beta } = \partial _x{\cal A}_y^{\alpha \beta }  \partial _y{\cal A}_x^{\alpha \beta } + i[ {{\cal A}_x,{\cal A}_y} ]^{\alpha \beta }\), in which α, β = 1, 2, ⋯, N. The Berry connection \({\cal A}_i^{\alpha \beta }({\bf{k}}) =  i\left\langle {u^\alpha ({\bf{k}})} \right{\textstyle{\partial \over {\partial k_i}}}\left {u^\beta ({\bf{k}})} \right\rangle\), where \(\left {u^{\alpha (\beta )}} \right\rangle\) are the periodic part of the Bloch wavefunctions (see ref. ^{1} for an introduction). Note that the trace of the commutator \({\mathrm{Tr}}[ {{\cal A}_x,{\cal A}_y} ]\) always vanishes for the first Chern class.
The topological invariants of our plainly modulated DG is C_{1} = (0, 0, 1). This can be understood from the original Weyl photonic crystal where \(C_1^z = 1\) for half of its Brillouin zone, as illustrated in Fig. 2b. By folding the Brillouin zone to half of its original size, the Chern numbers in different regions add up.
The 3D Chern crystal is a weak topological phase whose weak topological invariants are defined in a lower dimension, as compared with a strong topological phase with a strong topological invariant. It is theoretically known that a lattice dislocation in a weak topological phase creates a 1D topological defect mode^{23}. Unfortunately, in our case, a dislocation induces significant lattice distortion that generates many additional nontopological modes in the bandgap.
Fortunately, we propose and demonstrate below that, for a 3D Chern crystal constructed from Weyl crystals, a new approach is available: a smooth helical modulation generates a oneway mode at the core of the helix. The advantage of the helicalmodulation approach, compared with the latticedislocation approach, is the intactness of the lattice that prevents the generation of nontopological modes in the bandgap. We outline a physical interpretation as follows, and the rigorous calculations are presented in the Methods section. A supercell modulation couples two Weyl points of opposite chiralities, forming a gapped 3D Dirac point with a mass term that is complexvalued. (A 3D Dirac point consists of two Weyl points of opposite chiralities.) Then a helical modulation amounts to a nonzero winding number for the phase of the Dirac mass around the helical axis. It was indicated, in previous theoretical models, that such a topological perturbation can generate topological defect modes in both 2D^{24,25,26} and 3D systems^{27,28,29,30}.
Oneway fiber modes
Now comes the crucial step in our design of topological oneway fibers. Instead of the plain modulation (Fig. 2c), we create a helical modulation by filling the volume satisfying the inequality
The modulation now winds as a function of the angle θ [arctan(x, y)] in the x − y plane, whose spatial frequency is controlled by the signed integer w. The sign and magnitude of w determines the direction and number of the oneway modes on the winding axes. This is illustrated in the upper panels of Fig. 3 for w = +1, +2, +3, corresponding to single, double, and triple helix oneway fibers.
The band structures of the oneway fibers are shown in Fig. 3f, i, l. They were calculated using MIT Photonic Bands on a 11a × 11a × 2a cubic supercell. The spectra exhibit oneway modes within the bulk bandgap. The fields of the topological fiber modes are localized around the helix cores (Fig. 3g, j, m), and the localization length is minimized at the midgap frequencies. In general, the higherorder mode profiles are more extended in the real space. For the multimode fields of w = +2, +3, instead of the midgap frequencies, we plot the mode profiles close to the bandedge frequencies. Because multimode dispersions are almost degenerate at the middle of the bandgap (Fig. 3i, l), it is difficult to resolve their intrinsic mode patterns from their linear superpositions.
In Fig. 3f, i, l, all onewayfiber dispersions (green lines) have very similar phase and group velocities. In the multimode cases, their dispersions are almost degenerate at the midgap frequencies. This is due to the fact that these defect modes originated from the same Weyl bulk bands, so they all share the same Brillouinzone location and group velocities as those of the original Weyl cones. This behavior is different from that of the multimode oneway edge waveguides in 2D^{9}, where the edge modal dispersions have different phase or group velocities. This can be attributed to the fact that the edge environment, of sharp terminations, is distinct from the environment of the 2D bulk lattice. While, here, there are no sharp interfaces in the 3D oneway fibers. This unique feature, of multiple fiber modes having almost identical dispersions, ensures that multimode signals propagate at the same speed for both energy and phase.
Timedomain simulations
To visualize and confirm our prediction made by spectral dispersions, we simulate the wave dynamics of the oneway fiber (w = +1) in the real space in Fig. 4. Due to the huge computation domain, the finitedifference timedomain (FDTD) method is adopted for its nice scaling with the computation size. We use the commercial software EastWave^{31} for its capability in handling nonreciprocal materials.
In Fig. 4, we compare the oneway fiber to a regular fiber having a core diameter of 2a with a dielectric constant of 16 in air. In both cases, the computation domain is 20a × 20a × 26.5a, in (x, y, z) directions, and the mesh resolution is a/30. The perfectly matched layers (PMLs) are used at all six boundary planes. A dipole source polarized in z direction was placed at the position (0.1, 0.2, 5.5)a to excite the fiber mode. A metallic ball of diameter 1.5a is placed at (0, 0, 14.5)a to test the robustness of the mode. Obviously, the oneway mode perfectly circumvents the metal sphere without any scattering losses, while the regular fiber mode backscatters. We also note that fiber bends, disrupting the 3D bandgap of cladding, can cause photon loss.
Second Chern number
It is natural to ask for a topological invariant for the oneway fibers. With the simplest helix modulation of the form of Eq. (2), it is intuitive to guess that w is the topological invariant, since the number and direction of the oneway modes match the magnitude and sign of w. However, this observation does not work if we consider the modulation of the general form as \(f\left( {x,y,z} \right) \,> \,f_0 + \mathop {\sum}\nolimits_w {\kern 1pt} h_w{\kern 1pt} {\mathrm{cos}}\left( {\pi z{\mathrm{/}}a + w\theta } \right)\), where h_{w} are realvalued constants.
For a lattice dislocation in a 3D Chern crystal, it is known^{32} that the number of chiral modes is given by C_{1} · b, where the dimensionless Burgers vector (b) represents the magnitude and direction of the lattice distortion. However, this approach cannot be applied to our system due to the lack of a unique “Burgers vector” other than that in the simplest case (as of Eq. (2)).
We show that the formal topological invariant of our oneway fibers is the second Chern number (C_{2}), the strong topological invariant in our system. Note that, far away from the axis of the helix, the Bloch Hamiltonian smoothly varies with θ and is a smooth function of the four variables (k_{x}, k_{y}, k_{z}, θ). Since (k_{x}, k_{y}, k_{z}, θ) span a fourdimensional parameter space with periodic boundary conditions (a 4D torus), the second Chern number^{32,33} can be defined:
Similar to the definitions in Eq. (1), \({\cal F}_{ij}^{\alpha \beta } = \partial _i{\cal A}_j^{\alpha \beta }  \partial _j{\cal A}_i^{\alpha \beta } + i[ {{\cal A}_i,{\cal A}_j} ]^{\alpha \beta }\), in which α, β are the band indices. The nonAbelian Berry potential \({\cal A}_i^{\alpha \beta }({\bf{k}},\theta ) =  i\left\langle {u^\alpha ({\bf{k}},\theta )} \right\frac{\partial }{{\partial k_i}}\left {u^\beta ({\bf{k}},\theta )} \right\rangle\), where \(\left {u^{\alpha (\beta )}} \right\rangle\) are the eigenfunctions and k_{i} runs through k_{x}, k_{y}, k_{z}, θ. It is notable that this definition of C_{2} involves three variables (k_{x,y,z}) in the reciprocal space and one variable (θ) in the real space, in contrary to the four momentum variables in the 4D QHE^{33,34,35,36,37,38,39,40}. Consequently, the Berry curvature \({\cal F}_{i\theta }\) is even while \({\cal F}_{ij}\) is odd under time reversal, where i or j represents one of x, y, and z. Although in 4D QHE, C_{2} can be nonzero without breaking timereversal symmetry, nonzero C_{2} requires timereversal breaking in our system, which is consistent with the oneway phenomena.
In the Methods section, we carry out the explicit calculations of C_{2}, which is consistent with our numerical findings in Fig. 3. The topological protection by the second Chern number indicates that the physical origin of the oneway fiber modes is fundamentally different from that of the edge modes of the 2D Chern crystals^{5,6} (2D Chern insulator or 2D QHE), whose topology is captured by the first Chern number. We note that, although in our system both the weak indices (C_{1}) and the strong index (C_{2}) are nonzero, it is possible to construct a oneway fiber design with zero C_{1} and nonzero C_{2}. For example, when the separation between the two Weyl points shrinks to zero (forming a 3D Dirac point), one can apply only angular (θ) modulations to obtain a oneway fiber of nonzero C_{2} but zero C_{1}.
Discussion
Experimentally, oneway fibers can be constructed using gyromagnetic materials^{6,9,41} at microwave frequencies. For higher frequencies, there lacks magnetic materials with high Verdet constants and low loss. Nevertheless, we discuss the potential relevant technologies below. Toward optical frequencies, there is progress on magnetic fibers^{42,43} and gyroelectric materials^{44,45}. The optacoustic coupling^{46} in fibers provides another possibility for breaking timereversal symmetry. A DG fiber can either be made by drawing a 3Dprinted preform or potentially by selfassembly^{47,48} during the drawing process. Threedimensional direct writing^{49} and interference lithography^{50} can also be adopted. Finally, the chiral modulation can be created by spinning the fiber during drawing, as demonstrated in the chiral fibers^{51,52}.
The proposal of oneway fibers enriches the prospects of device applications for the Weyl materials and topological photonics. It also brings a new playground for the realization of higherdimensional topological physics^{53,54}. Topological fibers could also inspire new directions, design principles for novel fibers^{55,56,57}.
Methods
Effective 3D Dirac Hamiltonian
In the main text, we have presented our design of the oneway fiber and the results of bandstructure calculations. To gain a simple analytical understanding of the oneway modes, we outline below an effective Hamiltonian description, using the lowenergy Dirac Hamiltonian. The picture can be summarized as follows. In terms of the effective Dirac Hamiltonian, the modulation corresponds to the presence of a Dirac mass. The helix modulation introduces a topologically nontrivial configuration of the Dirac mass (nonzero winding number of the phase of Dirac mass), which creates topologically oneway defect modes.
In the absence of modulations, the doublegyroid crystal hosts two Weyl points with opposite chirality, either of which is described by a 2 × 2 effective Weyl Hamiltonian. We can combine them as a 4 × 4 blockdiagonal Dirac Hamiltonan: H_{D} = −iv(σ_{x}∂_{x} + σ_{y}∂_{y} + σ_{z}∂_{z})τ_{z} where σ_{i},τ_{i} (i = x, y, z) are Pauli matrices, v is the group velocity (for simplicity, we take isotropic group velocities with v > 0). As we have seen in the main text, a modulation with periodicity 2a in the z direction gaps out the Weyl points. In the effective DiracHamiltonian description, the frequency gap is due to the Dirac mass terms. There are only two possible Dirac mass terms: m_{1}τ_{x} and m_{2}τ_{y}, both of which anticommute with the σ_{i}τ_{z} terms in H_{D}. It is thus expected that the modulation amounts to the presence of these Dirac mass terms in the lowenergy effective Hamiltonian. In general, both m_{1} and m_{2} can be nonzero, and the mass term can be rewritten as m_{1}τ_{x} + m_{2}τ_{y} = mτ_{+} + m^{*}τ_{−}, with τ_{±} ≡ (τ_{x} ± iτ_{y})/2 and m ≡ m_{1} − im_{2}. The full effective Hamiltonian, with the effect of modulation included as the Dirac mass, can be written as
from which we can readily see that a frequency gap 2m is generated by the Dirac mass terms.
In this effective Hamiltonian, the chirality flipping terms containing τ_{±} couple the states near the two Weyl points; therefore, they should come from the modulation. Suppose that the modulation can be modeled in the effective potential \(V({\bf{r}}) = V_{\bf{Q}}{\kern 1pt} {\mathrm{exp}}\left( { + i{\bf{Q}} \cdot {\bf{r}}} \right) + V_{\bf{Q}}^\dagger {\mathrm{exp}}\left( {  i{\bf{Q}} \cdot {\bf{r}}} \right) + \cdots\), where Q = (0, 0, π/a) is the wavevector that couples the two Weyl points. We can see that exp(±iQ · r) → τ_{±} is valid near the Weyl points, and we have m = V_{Q}, in other words, the complexvalued Dirac mass is simply the Qcomponent of the perturbation.
Now, we show that the displacement of modulation (d) amounts to the phase change of the Dirac mass (m). If we displace the modulation by a distance d, then the perturbation becomes V(r + d), which can be expanded as V(r + d) = \(V_{\bf{Q}}{\mathrm{exp}}\left[ {i{\bf{Q}} \cdot ({\bf{r}} + {\bf{d}})} \right] + V_{\bf{Q}}^\dagger {\mathrm{exp}}\left[ {  i{\bf{Q}} \cdot ({\bf{r}} + {\bf{d}})} \right] + \cdots\), thus we can see that the displacement causes V_{Q} → V_{Q} exp(iQ · d), or equivalently, m → m exp(iQ · d). For the helixshape modulation along the axis r = 0, as described in the main text, the displacement d is a function of θ such that Q · d = wθ. Here, we use the cylindrical coordinates (x, y, z) ≡ (rcosθ, rsinθ, z). Therefore, we have a nonzero winding of the phase of Dirac mass around the axis, namely, m(θ) = m_{0} exp(iwθ), in which m_{0} ≡ m(θ = 0). The overall phase of m_{0} can be changed by rotating the coordinate systems around the r = 0 axis, thus we are free to take m_{0} to be realvalued and positive.
Analytic solutions of oneway modes
For the effective 3D Dirac Hamiltonian with a nonzero winding of the phase of Dirac mass (with winding number w), which is a consequence of the helix perturbation, we shall show that there exist w topological oneway modes. We can rewrite Eq. (4) in the cylindrical coordinates as
where we have taken advantage of the translational symmetry in the z direction by replacing −i∂_{z} by k_{z}. For notational simplicity, we shall keep implicit the common factor exp(ik_{z}z) in the eigenfunction. For a reason that will become clear shortly, we look for eigenfunctions of the form of ψ = [ψ_{1}, 0, 0, ψ_{4}]^{T}. The eigenvalues are E = vk_{z}, and the eigenfunctions satisfy
It is not difficult to observe that the second equation is equivalent to the first one if we take \(\psi _4 = \pm \psi _1^ \ast\). Let us focus on the \(\psi _4 = + \psi _1^ \ast\) case first. With this condition, the above two equations are reduced to a single equation
For the w = +1 case, the common exp(iθ) factors can be eliminated; thus, the equation becomes especially simple, and the solution can be found analytically as
For an arbitrary integer w ≥ +1, by analysis analogous to ref. ^{24}, we can show that there exist w localized modes. In fact, we can take the following ansatz for Eq. (7):
with an integer parameter l, whose acceptable values are to be determined. We first notice that when l = (w − 1)/2, e^{ilθ} and e^{i(w−1−l)θ} are actually equal, and the v_{l} term is redundant. Let us first focus on the cases l ≠ (w − 1)/2. The special case l = (w − 1)/2, with e^{ilθ} = e^{i(w−1−l)θ}, will be discussed separately later.
According to Eq. (7), the coefficient functions u_{l},v_{l} have to satisfy
The asymptotic behaviors of u_{l}, v_{l} in the r → 0 limit can be found as (I) u_{l} : r^{l}, v_{l} ~ r^{l+1} or (II) u_{l} : r^{w−l}, v_{l} : r^{w−l−1}. On the other hand, the asymptotic behaviors in the r → ∞ limit are (a) \(u_l \to {\mathrm{exp}}\left( {  \frac{{m_0}}{v}r} \right),v_l \to {\mathrm{exp}}\left( {  \frac{{m_0}}{v}r} \right)\) or (b) \(u_l \to {\mathrm{exp}}\left( {\frac{{m_0}}{v}r} \right),v_l \to {\mathrm{exp}}\left( {\frac{{m_0}}{v}r} \right)\), only the first of which is normalizable in the r → ∞ regime. A normalizable solution must have behavior (a) in the r → ∞ limit, which is generally a superposition of (I) and (II) in the r → 0 regime. Therefore, the normalizability of the solution in the r → 0 limit requires that both (I) and (II) are normalizable, which leads to the constraint
Thus, we have proved that, leaving out the special case l ≠ (w − 1)/2 undetermined, there exists one normalizable solution for every integer l = 0, 1, 2, ⋯, w − 1. However, the solutions with l > (w − 1)/2 are redundant, because the solutions for l and l′ with l + l′ = w −1 are actually the same one, as can be appreciated from Eq. (9). Therefore, the total number of solutions with \(\psi _4 = + \psi _1^ \ast\) is the number of a nonnegative integer smaller than (w − 1)/2, which is [w/2] (here, “[⋯]” denotes the floor function, mapping a real number to the largest previous integer), excluding a possible solution with l = (w − 1)/2.
Now we consider the other choice: \(\psi _4 =  \psi _1^ \ast\). By calculations similar to the case \(\psi _4 = \psi _1^ \ast\), we can obtain equations similar to Eq. (7), except that the “+” sign before m_{0} is replaced by “−”. We adopt the same ansatz as given in Eq. (9), and follows the steps below Eq. (9), solving the case l ≠ (w − 1)/2 first. It is found that the number of solutions is [w/2].
Finally, we study the special case l = (w − 1)/2 (this case needs consideration only when w is odd; for w even, this option is automatically absent). Given this value of l, the second term in Eq. (9) becomes redundant, thus we can take
Under the choice \(\psi _4 = \pm \psi _1^ \ast\), we obtain the single differential equation
For the choice “+” of the “±”, Eq. (13) has a single normalizable solution with asymptotic behaviors u_{l} → r^{l} in the r → 0 limit and \(u_l \to {\mathrm{exp}}\left( {  \frac{{m_0}}{v}r} \right)\) in the r → ∞ limit. For the choice “−” of the “±”, Eq. (13) leads to \(u_l \to {\mathrm{exp}}\left( {\frac{{m_0}}{v}r} \right)\) in the r → ∞ limit, which is apparently not normalizable. Therefore, there exists a single normalizable localized mode, in the \(\psi _4 = + \psi _1^ \ast\) sector, for the special case l = (w − 1)/2. We also note that, if we take m_{0} < 0 instead of m_{0} > 0, the normalizable solution would be present in the \(\psi _4 =  \psi _1^ \ast\) sector but absent in the \(\psi _4 = + \psi _1^ \ast\) sector, thus the total number of solution is the same.
Let us summarize the above calculations as follows. When w is odd, the total number of normalizable solutions is 2[w/2] + 1 = w; when w is even, the total number of normalizable solutions is 2[w/2] = w. Therefore, the total number of topological modes is always w, irrespective of the parity (odd/even) of w. Furthermore, our calculation shows that the eigenvalues take the simple form
Thus, all these w modes propagate along the +z direction, with the same velocity v.
Finally, let us discuss the oneway modes for the integer w < 0. A solution of the form of ψ = [ψ_{1}, 0, 0, ψ_{4}]^{T} does not exist in this case, because the condition given in Eq. (11) can never be satisfied. On the other hand, solutions of the form of ψ = [0, ψ_{2}, ψ_{3}, 0]^{T} can be found. In fact, we can follow the steps above and obtain the equations
whose complex conjugations are
We can see that Eq. (16) is the same as Eq. (6) except that ψ_{2} and −ψ_{3} take the place of ψ_{1} and ψ_{4}, respectively. Now, our previous analysis for Eq. (6) with w ≥ + 1 immediately tells us that the number of oneway modes for w < 0 is w. Because the solutions for w < 0 take the form of ψ = [0, ψ_{2}, ψ_{3}, 0]^{T}, the dispersion is E(k_{z}) = −vk_{z}, thus all the oneway modes propagate in the −z direction.
Calculation of the second Chern number C _{2}
The effective Hamiltonian Eq. (4) takes the form of
where the Dirac matrices Γ^{a} = σ_{a}τ_{z} (a = 1, 2, 3), Γ^{4} = τ_{x}, Γ^{5} = τ_{y}, the coefficient functions d_{a} = vk_{a} (a = 1, 2, 3), d_{4} = Re(m), d_{5} = −Im(m). A straightforward calculation of C_{2}, as in ref. ^{33}, leads to
where \(\hat d_a = d_a{\mathrm{/}}\sqrt {\mathop {\sum}\nolimits_{b = 1}^5 d_b^2}\). With the Dirac mass m = m_{0} exp(iwθ), we have
For a general modulation that combines several different spatial frequencies, namely, \(m(\theta ) = \mathop {\sum}\nolimits_w {\kern 1pt} m_w{\mathrm{exp}}(iw\theta )\), Eq. (18) is not amenable to further simplification in the generic cases. However, we have C_{2} = w_{0} in the cases that \( {m_{w_0}}  \,> \mathop {\sum}\nolimits_{w \ne w_0}  {m_w} \), in other words, C_{2} is determined by the dominant modulation.
Data availability
All relevant data are available from the authors on request.
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Acknowledgements
We thank the discussion with Jian Wang, Wei Ding, and Changyuan Yu on fiber technologies and with Hannah M. Price and Chen Fang on 3D QHE. L.L. was supported by the National key R&D Program of China under grant no. 2017YFA0303800, 2016YFA0302400, and by NSFC under project no. 11721404. Z.W. was supported by NSFC under grant no. 11674189.
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L.L. and Z.W. initiated the project and prepared the manuscript. L.L. computed the band structures, H.G. performed the timedomain simulations, and Z.W. obtained the analytical solutions.
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Lu, L., Gao, H. & Wang, Z. Topological oneway fiber of second Chern number. Nat Commun 9, 5384 (2018). https://doi.org/10.1038/s41467018078173
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DOI: https://doi.org/10.1038/s41467018078173
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