Introduction

Topological matters have garnered broad ramifications in photonics, acoustics and mechanicals, because of the existence of topological states with robustness or immunity to backscattering and disorder1,2,3,4,5,6,7. Particularly, higher-order topological matters hallmarked by hierarchical topological states (HTSs) with multiple dimensions, have emerged as a pioneering concept for manipulating waves in integrated applications such as lasers, routers and filters for information processing, computing and sensing8,9,10,11, with the first experimental verification in microwave systems12. Understanding the bulk-boundary correspondence is essential for predicting and implementing these exotic physical phenomena, by which topological behaviors must be founded upon bulk lattices13,14,15,16. However, this in turn limits the flexibility of controlling waves and imposes large-scale period requirements of bulk lattices.

Non-Hermiticity naturally making spectra of a system from real to complex, can be employed for describing non-conservative systems17,18,19,20,21. It arises from the energy exchange between physical systems and open environments, reflected by the presence of gain and loss, and non-reciprocity22,23,24,25,26. Non-Hermitian skin effect (NHSE), a representative behavior under non-Hermitian formalism, refers to wave or particle confinement toward open boundaries for an extensive number of bulk states, with contrasting implications on fundamental bulk-boundary correspondence25,27,28,29,30,31. It suggests the failure of Bloch’s theorem and ignites extensive interest in wave systems such as quantum walks32,33,34, acoustics35,36,37, and electric circuits38,39,40, inspiring further studies on higher dimensional non-Hermitian generalizations25,31,41,42,43,44,45. Despite these advances, one of the most relied upon cornerstones of these NHSEs is the bulk state in a generalized Brillouin zone46,47,48. Recently, the coexistence of NHSEs and topological states has been identified theoretically and experimentally, yet the interplay mechanism of NHSEs and HTSs remains unclear26,49,50,51,52.

Here, we present the non-Hermitian reconstruction of HTSs by inducing the interplay of NHSEs and HTSs. A two-dimensional (2D) kagome lattice is introduced in our theoretical framework, in which HTSs are spatially redistributed through precise manipulation of non-reciprocal couplings. The mechanism of forming complex spectra of HTSs is revealed by a non-reciprocal coupling framework and a macroscopic non-Hermitian domain wall model. Specifically, because of the requirement for spatially non-negligible couplings, zero-dimensional (0D) topological states of HTSs only undergo non-Hermitian reconstruction when the system size is finitely small. While 2D topological states of HTSs must undergo non-Hermitian reconstruction as long as the edge exists. Although these HTSs become reconstructed under the NHSE, their spectra remain isolated within the bulk bandgap, exhibiting persistent topological protections. A photonic platform is implemented by coupled resonator optical waveguides (CROWs) with gain and loss to mimic these physical phenomena53,54,55. Our findings suggest flexibly reconstructable HTSs under the incorporation of NHSEs and HTSs, which go beyond the scope of previous understanding toward non-Hermitian topology and benefit a set of potential topological applications.

Results

Non-Hermitian topological lattices with HTSs

HTSs imply the coexistence of multidimensional topological states in a system. To construct non-Hermitian HTSs, a 2D kagome lattice with non-reciprocal couplings is considered, as illustrated in Fig. 1a. There are three directions of interactions that can be controlled independently, labeled by α = 1, 2, 3. The bulk Hamiltonian is written as:

$$H={H}_{{{{\rm{intra}}}}}+{H}_{{{{\rm{inter}}}}},$$
(1)

where Hintra = ∑λ = ±Hλ with \({H}^{-}={\sum }_{i,j}({t}_{1}^{-}{A}_{i,j}^{{{\dagger}} }{B}_{i,j}+{t}_{2}^{-}{B}_{i,j}^{{{\dagger}} }{C}_{i,j}+{t}_{3}^{-}{C}_{i,j}^{{{\dagger}} }{A}_{i,j})\) and \({H}^{+}={\sum }_{i,j}({t}_{1}^{+}{B}_{i,j}^{{{\dagger}} }{A}_{i,j}+{t}_{2}^{+}{C}_{i,j}^{{{\dagger}} }{B}_{i,j}+{t}_{3}^{+}{A}_{i,j}^{{{\dagger}} }{C}_{i,j})\), and \({H}_{{{{\rm{inter}}}}}={t}_{0}{\sum }_{i,j}({A}_{i,j}^{{{\dagger}} }{B}_{i,j-1}+\,{A}_{i,j}^{{{\dagger}} }{C}_{i-1,j}+{B}_{i,j}^{{{\dagger}} }{A}_{i,j+1}+{B}_{i,j}^{{{\dagger}} }{C}_{i-1,j+1}+{C}_{i,j}^{{{\dagger}} }{A}_{i+1,j}+{C}_{i,j}^{{{\dagger}} }{B}_{i+1,j-1}+H.c.)\). \({{{\Lambda }}}_{i,j}^{({{\dagger}} )}\) denotes the annihilation (creation) operator for sublattice sites (Λ = A, B, C) at position (i, j). \({t}_{\alpha }^{\pm }=t\pm \delta {t}_{\alpha }\) and t0 indicate the intracell and intercell couplings, respectively. Non-Hermiticity in Eq. (1) originates from the inequivalence between \({t}_{\alpha }^{-}\) and \({t}_{\alpha }^{+}\), as depicted in Fig. 1b (see Supplementary Note 1).

Fig. 1: Lattice model with hierarchical topological states (HTSs) and 0D non-Hermitian reconstruction of HTSs.
figure 1

a Sketch of the kagome lattice model with finite sizes and non-reciprocal couplings. b Unit cell consisting of three sites with inter-cell couplings (t0) and intra-cell non-reciprocal couplings (\({t}_{\alpha }^{-}\) and \({t}_{\alpha }^{+}\)). c Two equivalent Brillouin zones in our lattices. d Spectrum for unit cells under periodic boundary conditions (PBCs) and the spectrum for (a) under open boundary conditions (OBCs) (N = 10), with t0 = 1.5 and t = 0.5. e Maximum local density of states (LDOSs) for 0D Hermitian topological states and 0D non-Hermitian topological states. The first panel shows the Hermitian scenario with ∀ δtα = 0. The second to fourth panels show the non-Hermitian scenarios with topological localizations at different numbers of corners, e.g., δt1 > 0 and δt2 < 0 (one corner), δt1 > 0 and δt3 < 0 (two corners) as well as ∀ δtα > 0 (three corners).

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Since small δti has a negligible effect on phase transitions, the topology of our system can be characterized by a generalized 2D non-Hermitian bulk polarization associated with the Wannier centers, that is, the center of maximally localized Wannier functions (see Supplementary Note 1). It is defined as \({P}_{\rho }=-\frac{1}{{(2\pi )}^{2}}{\int}_{{{{{{\rm{BZ}}}}}}}{{{{{\rm{Tr}}}}}}[{{{{{{\mathcal{A}}}}}}}_{\rho }]{d}^{2}{{{{{\bf{k}}}}}}\), ρ = 1, 2, where \({({{{{{{\mathcal{A}}}}}}}_{\rho })}_{mn}({{{{{\bf{k}}}}}})=i\langle {u}_{m,{{{{{\bf{k}}}}}}}^{L}\vert {\partial }_{{k}_{\rho }}\vert {u}_{n,{{{{{\bf{k}}}}}}}^{R}\rangle \), with \(\langle {u}_{m,{{{{{\bf{k}}}}}}}^{L}\vert \) and \(\vert {u}_{n,{{{{{\bf{k}}}}}}}^{R}\rangle \) being mth left and nth right eigenvectors (below the considered band gap), respectively. ρ refers to two distinct directions marked in Fig. 1c. Exact calculation can be performed by a Wilson-loop approach \({P}_{{\rho }_{1}}=\frac{1}{2\pi }{\int}_{{{{{{{\rm{BZ}}}}}}}_{{\rho }_{2}}}d{v}_{{k}_{{\rho }_{2}}}^{{\rho }_{1}}\), where \({v}_{{k}_{{\rho }_{2}}}^{{\rho }_{1}}\) is the Berry phase along the closed loop \({k}_{{\rho }_{1}}\) for fixed \({k}_{{\rho }_{2}}\), and ρ1,2 represent two different ρ choices. Maintaining all elements of P for bulk bands nonzero guarantees the presence of HTSs. This indicates that Wannier centers coincide with Wyckoff positions. In the Hermitian limit (∀ δtα = 0), the system is trivial for t > t0 (Pρ = 0) and topologically nontrivial for t < t0 (∣Pρ∣ = 1/3). In Fig. 1d, we plot the spectra for nontrivial cases subject to periodic boundary conditions (PBCs) and open boundary conditions (OBCs). Isolated 0D and 1D topological states exist simultaneously within the bulk bandgap. In the non-Hermitian limit (∃ δtα ≠ 0), however, the eigenenergy of Eq. (1) is generally complex \(E\in {\mathbb{C}}\), and this makes the emergence of NHSEs for bulk states possible under OBCs (see Supplementary Note 2). Nevertheless, our definition of P is still available, i.e., ∣Pρ∣ = 1/3 holds for nontrivial cases and ∣Pρ∣ = 0 holds for trivial cases. Consequently, non-Hermitian HTSs can exist stably in our system. Examining detailed properties for non-Hermitian HTSs relies on specific analysis of NHSEs in the following.

0D non-Hermitian reconstruction of HTSs

Ideally, 0D topological states of HTSs are fixed at zero energy owing to chiral symmetry in this system. This requires a large number of periodic lattices, consequently, multiple 0D topological states of HTSs almost do not overlap with each other. However, at finitely small sizes of triangular regions in this system, multiple 0D topological states of HTSs must yield inevitable redistribution through energy exchange. A local density of states (LDOS) in real space can be provided to capture this feature:

$$f({E}_{},{{{{{\bf{r}}}}}})=\mathop{\sum}\limits_{\eta }| {\psi }_{\eta }({{{{{\bf{r}}}}}}){| }^{2}\delta (E-{E}_{\eta }),$$
(2)

where ψη(r) is the ηth normalized eigenvector with OBCs, and r stands for position coordinates. Consider the 0D non-Hermitian reconstruction of HTSs with regard to E → 0, as shown in Fig. 1e. When ∀ δtα = 0, the maximum LDOS is localized at three corners in a reciprocal coupling framework (see Supplementary Note 3). When ∃ δtα ≠ 0, three scenarios come into view. By controlling δtα in signs and absolute values, the maximum LDOS is localized at the corner where the energy flow goes, with their numbers being one, two or three. This can be described qualitatively in a non-reciprocal coupling framework (see Supplementary Note 3). The flexibly tunable non-Hermitian behavior toward 0D topological states of HTSs is built upon finite size effects and stays robust against local disorders56,57.

1D non-Hermitian reconstruction of HTSs

1D topological states of HTSs are distributed at 1D closed boundaries, showing exponential decay in bulks. As such, their non-Hermitian reconstruction can be understood by a macroscopic model of non-Hermitian multiple domain walls with connected heads and tails, as illustrated in Fig. 2a. Boundary sites exhibit dimeric configuration naturally, with l from 1 to 3N indexing their lattices clockwise, so the normalized boundary eigenvector is given by \(\left\vert {{\Psi }}\right\rangle ={({{{\Phi }}}_{1}^{{{{{{\rm{T}}}}}}},...,{{{\Phi }}}_{3N}^{{{{{{\rm{T}}}}}}})}^{{{{{{\rm{T}}}}}}}\). At these boundaries, we have the recurrence relation derived from the time-independent Schrödinger equation:

$${{{\Gamma }}}_{1}{{{\Phi }}}_{l-1}+{{{\Gamma }}}_{2}{{{\Phi }}}_{l}+{{{\Gamma }}}_{1}^{{{{{{\rm{T}}}}}}}{{{\Phi }}}_{l+1}=E{{{\Phi }}}_{l},$$
(3)

where Γ1 = (0, t0; 0, 0) and \({{{\Gamma }}}_{2}=(0,{t}_{\alpha }^{-};{t}_{\alpha }^{+},0)\). The general solution for Eq. (3) takes the form \({{{\Phi }}}_{l}={({\phi }_{l,a},{\phi }_{l,b})}^{{{{{{\rm{T}}}}}}}=\mathop{\sum }\nolimits_{q = 1}^{2}{z}_{q,\alpha }^{l}{({\varphi }_{q,\alpha ,a},{\varphi }_{q,\alpha ,b})}^{{{{{{\rm{T}}}}}}}\), with remarked sublattices a and b (see “Methods”). Substituting this expression into Eq. (3), we obtain the characteristic equation for nonzero φq,α,a and φq,α,b:

$$\begin{array}{l}{z}_{q,\alpha }^{2}{t}_{\alpha }^{-}{t}_{0}+{z}_{q,\alpha }({t}_{\alpha }^{+}{t}_{\alpha }^{-}-{E}^{2})+{t}_{\alpha }^{+}{t}_{0}=0,\end{array}$$
(4)

here zq,α provides the link between Φl and E, which is crucial for constructing non-Bloch Brillouin zones. Assuming that ∣z1,α∣ > ∣z2,α∣, ∣zμ∣ can be defined for the μth largest value among eight terms \(| {z}_{{q}_{1},1}{z}_{{q}_{2},2}{z}_{{q}_{3},3}| \), where q1, q2, q3 ∈ {1, 2}. Then we impose corner conditions ϕ1,a = ϕ3N,b, ϕN+1,a = ϕN,b and ϕ2N+1,a = ϕ2N,b on Eq. (3), and derive non-Hermitian domain wall conditions with respect to \(g=\frac{| {t}_{1}^{+}{t}_{2}^{+}{t}_{3}^{+}| }{| {t}_{1}^{-}{t}_{2}^{-}{t}_{3}^{-}| }\), as displayed in Fig. 2b (see “Methods”).

Fig. 2: Macroscopic model of non-Hermitian multiple domain walls (1D topological states).
figure 2

a Schematic diagram of closed non-Hermitian multiple domain walls. The non-reciprocity of three domain walls can be described by (\({t}_{1}^{-},{t}_{1}^{+}\)), (\({t}_{2}^{-},{t}_{2}^{+}\)) and (\({t}_{3}^{-},{t}_{3}^{+}\)), respectively. b Conditions for valid solutions of this macroscopic model, which are categorized into three cases (I–III). g is defined as \(\frac{| {t}_{1}^{+}{t}_{2}^{+}{t}_{3}^{+}| }{| {t}_{1}^{-}{t}_{2}^{-}{t}_{3}^{-}| }\) and ∣z1(2)∣ represents the (second) largest absolute value of the product of complex wavevectors on three boundaries. c Spectra, zq,α and boundary field distributions for different cases. Here we have δt1 = δt3 = 0.1 and δt2 = 0.2 for case I, δt1 = −δt3 = 0.1 and δt2 = 0 for case II as well as δt1 = −0.1 and δt2 = δt3 = −0.2 for case III. Other parameters are t0 = 1.5 and t = 0.5. The TBM calculation is based on N = 50. The black dotted lines are closed loops for ideally ∣zq,α∣ = 1. The red, green and blue curves indicate allowed complex wavevectors on three boundaries, corresponding to α = 1, 2, 3, respectively. a1, a3 and a5 indicate 0D non-Hermitian topological states, which are not significantly reconstructed because the array is not small enough. a2 and a6 indicate 1D non-Hermitian topological states with maximum imaginary parts of eigenenergies. a4 indicate 1D non-Hermitian topological states with E = 1.5.

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The field distribution and spectral information of 1D non-Hermitian topological states become accessible theoretically for a given g. In Fig. 2c, we plot the cases g > 1 (I), g = 1 (II) and g < 1 (III). The spectra obtained by the macroscopic model are consistent with those calculated from the complete tight-binding model (TBM). zq,α in complex planes encloses the field distribution information of 1D non-Hermitian topological states. For the left panels of Fig. 2c, ∣zq,2∣ > 1 almost invariably holds, indicating exponential accumulation at the lower right corner. The middle panels of Fig. 2c suggest ∣zq,1∣ > 1, ∣zq,2∣ = 1 and ∣zq,3∣ < 1, which imply exponential accumulation at the upper middle and lower right corners and a standing wave formed by Bloch eigenvectors at the upper right boundary. For the right panels of Fig. 2c, ∣zq,2∣ < 1 and ∣zq,3∣ < 1 almost invariably hold, indicating exponential accumulation at the upper middle corner. Our analysis not only verifies the effective description of the macroscopic model but also shows abundant possibilities of 1D non-Hermitian reconstruction of HTSs.

Scaling behavior of 1D non-Hermitian reconstruction

A hallmark of NHSEs involved in the 1D non-Hermitian reconstruction of HTSs is the scale-free behavior, particularly in eigenstates and eigenenergies. Since the presence of ∣zq,α∣ ≠ 1, 1D non-Hermitian reconstructed topological states contain exponentially enhancing or decaying factors apart from Bloch functions, i.e., ∣Ψ(x)∣ ~ eκx and ∣κ∣ ~ N−1, without fixed length scale. Figure 3a as a validation illustrates the eigenvectors (a2) in the left panels of Fig. 2c at different system sizes N = 30, 40, 50, and 60, with the horizontal axis normalized by the site number at three boundaries 6N − 3. Meanwhile, the eigenenergies with extreme imaginary parts for particular boundary size N satisfy a qualitative relation with that for infinite boundary sizes, i.e., \({E}_{{{{{{\rm{OBC}}}}}}}^{+\infty }-{E}_{{{{{{\rm{OBC}}}}}}}^{N} \sim {N}^{-1}\), as shown in Fig. 3b. Such an exotic scale-free behavior results from the asymptotic solutions when the system scales from a finite size to infinity.

Fig. 3: Scaling behavior of 1D non-Hermitian reconstruction.
figure 3

a Boundary field distributions for a2 with N = 30, 40, 50, 60. The orange arrows label the trend of field distributions as N increases and the dashed lines denote the corner positions. b Difference between eigenenergies with maximum imaginary parts for N → +∞ and a given N, which is proportional to N−1. The TBM calculation is consistent with the fitting result.

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CROW implementation

Achieving the interplay of NHSEs and HTSs crucially depends on controllable non-reciprocal coupling, which presents a challenge in photonics. CROWs can introduce effective gauge fields without breaking time-reversal symmetry and become an efficient platform for examining non-reciprocal physics. Every unit cell of our CROW system (2D kagome lattices) consists of three whispering gallery resonator sites that are coupled with each other through link resonators, as shown in Fig. 4a. These resonators have a higher refractive index than the surrounding air (e.g., the silicon in our proposal, n = 3.48) and therefore support both clockwise and anti-clockwise modes (the emulation of spin degrees of freedom), as illustrated in Fig. 4b. Intra- and inter-cell couplings are determined by distances between site resonators and link resonators (d1 and d2) as well as the gain/loss configuration of link resonators. Without applying any gain and loss, clockwise and anti-clockwise modes share the same coupling parameters. HTSs appear under OBCs when d1 > d2, with their spectra shown in Fig. 4c, where each blue dot represents two spin-degenerate modes. In contrast, the d1 < d2 condition makes inter-cell couplings weaker than intra-cell couplings and hence gives rise to a trivial insulator.

Fig. 4: Designed CROW system supporting the non-Hermitian reconstruction of HTSs.
figure 4

a Unit cell of the designed CROW system, where d1 (d2) represents the distance between intra-cell (inter-cell) link resonators and site resonators. Link resonators have an inner ring radius rc and a straight waveguide length L (upper inset), which support the refractive index modulation for two half regions (n ± iκα). Site resonators have an inner ring radius r0 (bottom inset). All these waveguides have the same width W. b Two spin polarizations for THSs enabling opposite propagation trajectories. c Complex spectra for different imaginary parts of refractive indices. Blue dots have the parameters κ1 = κ2 = κ3 = 0, while red dots have the parameters κ1 = κ3 = 0.04 and κ2 = 0.06. Other parameters are set as N = 10, r0 = 1.8 μm, rc = 1 μm, L = 1.54 μm, d1 = 0.16 μm, d2 = 0.04 μm and W = 0.2 μm. d 0D reconstructed topological state distributed at three corners (labeled as X in c). e 1D reconstructed topological state with respect to two spin polarizations (labeled as Y in c). f Reconstructed bulk state in the absence of NHSEs (labeled as Z in c). Blue arrows from (d) to (f) mark the directions of field accumulation.

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To realize the non-Hermiticity for HTSs, intra-cell link resonators are carefully designed with half gain and half loss configurations for different spin polarizations (imaginary parts of refractive indices ∃ κα ≠ 0), as displayed in Fig. 4a. This can be mimicked by placing metals on these resonators to produce biased optical losses28. Each spin polarization corresponds to one type of non-reciprocal couplings, while two spin polarizations appear in pairs having opposite non-reciprocal couplings. Red dots of Fig. 4c show the complex spectra for the aforementioned coexistence of g > 1 and g < 1. Here, because of ∀ κα ≠ 0 the non-Hermitian reconstruction of 0D topological states makes their eigenfrequencies lie near 192.9 THz of complex spectra (the azimuthal mode number is 21), with fields distributed at three corners (Fig. 4d), exhibiting the features of the rightmost panel of Fig. 1e. The non-Hermitian reconstruction of 1D topological states yields generalized topological NHSEs for two spin polarizations simultaneously (Fig. 4e). This is consistent with the results predicted by our theoretical model both in complex spectra and eigenstates (Fig. 2c). Meanwhile, contrary to previous higher-order NHSEs, bulk states in our system do not necessarily involve NHSEs even though non-reciprocal couplings occur (Fig. 4f)25. All these phenomena can be captured by accurate numerical simulations (see “Methods” and Supplementary Note 4).

Discussion

The interplay of NHSEs and HTSs has been adequately elucidated by a non-reciprocal coupling framework and a macroscopic model, exhibiting non-Hermitian multidimensional topological states with controllable field distributions. We show that finitely small system sizes are necessary for achieving the reconstruction of all HTSs (primarily limited by 0D topological states), which impose higher requirements than general NHSEs. This reconstruction remains protected by higher-order topology with robustness against local disorders and is fundamentally different from general higher-order NHSEs where low-dimensional localizations appear for an extensive number of bulk states (see Supplementary Note 5)25,31. These analyses provide an efficient non-Hermitian framework for modeling interactions among multiple 0D topological states and a theoretical paradigm for acquiring exact solutions of generic non-Hermitian domain wall problems48. By introducing spin-dependent gain/loss modulation to link resonators of CROW systems, a feasible photonic platform with non-reciprocity is constructed to demonstrate our theoretical predictions, which also enables extensive studies of non-Hermitian effects, spinful physics and integrated technologies. Specifically, reconstructed 0D topological states studied here support individual alterations both in number and orientation, available for tunable topological photonic lasers55,58. Meanwhile, reconstructed 1D topological states enable propagation along specific boundary pathways and are expected to be applied in photonic routing23. At these corners and pathways, the field accumulation can take place, offering potential applications for photonic funneling59. In light of the flexibility of this reconstruction, the model in this work can be generalized to suit different photonic needs45,49. Our findings open new grounds for non-Hermitian topological manipulations and readily promise a transfer to other systems that support non-reciprocal couplings (although validated in photonics), e.g., acoustics25,35,36, electric circuits38,39,40 and cold atoms49.

Methods

Non-Hermitian multiple domain wall model

In our system, 1D topological states of HTSs can be treated approximately as bulk states of non-Hermitian multiple domain walls with connected heads and tails. For each non-Hermitian domain wall (α = 1, 2, 3), we have:

$$\left\{\begin{array}{l}E{\phi }_{l,a}={t}_{0}{\phi }_{l-1,b}+{t}_{\alpha }^{-}{\phi }_{l,b},\\ E{\phi }_{l,b}={t}_{0}{\phi }_{l+1,a}+{t}_{\alpha }^{+}{\phi }_{l,a},\end{array}\right.$$
(5)

where \({\phi }_{l,a}=\mathop{\sum }\nolimits_{q = 1}^{l}{z}_{q,\alpha }^{l}{\varphi }_{q,\alpha ,a}\), \({\phi }_{l,b}=\mathop{\sum }\nolimits_{q = 1}^{2}{z}_{q,\alpha }^{l}{\varphi }_{q,\alpha ,b}\) and \({{{\Phi }}}_{l}={({\phi }_{l,a},{\phi }_{l,b})}^{{{{{{\rm{T}}}}}}}\). Using the conditions of φq,α,a ≠ 0 and φq,α,b ≠ 0, one can easily obtain Eq. (4). The corner conditions ϕ1,a = ϕ3N,b, ϕN+1,a = ϕN,b and ϕ2N+1,a = ϕ2N,b can be further expanded, given by:

$$\begin{array}{l}MV=0,\end{array}$$
(6)

where \(V={({\varphi }_{1,1,a},{\varphi }_{2,1,a},{\varphi }_{1,2,a},{\varphi }_{2,2,a},{\varphi }_{1,3,a},{\varphi }_{2,3,a})}^{{{{{{\rm{T}}}}}}}\) and the coefficient matrix is written as:

$$M=\left[\begin{array}{cccccc}{W}_{1,1}{t}_{0}&{W}_{2,1}{t}_{0}&0&0&-{z}_{1,3}^{3N}{t}_{3}^{+}&-{z}_{2,3}^{3N}{t}_{3}^{+}\\ {z}_{1,1}{W}_{1,1}{t}_{1}^{-}&{z}_{2,1}{W}_{2,1}{t}_{1}^{-}&0&0&-{z}_{1,3}^{3N+1}{t}_{0}&-{z}_{2,3}^{3N+1}{t}_{0}\\ {z}_{1,1}^{N+1}{t}_{0}&{z}_{2,1}^{N+1}{t}_{0}&-{z}_{1,2}^{N+1}{W}_{1,2}{t}_{2}^{-}&-{z}_{2,2}^{N+1}{W}_{2,2}{t}_{2}^{-}&0&0\\ {z}_{1,1}^{N}{t}_{1}^{+}&{z}_{2,1}^{N}{t}_{1}^{+}&-{z}_{1,2}^{N}{W}_{1,2}{t}_{0}&-{z}_{2,2}^{N}{W}_{2,2}{t}_{0}&0&0\\ 0&0&{z}_{1,2}^{2N+1}{t}_{0}&{z}_{2,2}^{2N+1}{t}_{0}&-{z}_{1,3}^{2N+1}{W}_{1,3}{t}_{3}^{-}&-{z}_{2,3}^{2N+1}{W}_{2,3}{t}_{3}^{-}\\ 0&0&{z}_{1,2}^{2N}{t}_{2}^{+}&{z}_{2,2}^{2N}{t}_{2}^{+}&-{z}_{1,3}^{2N}{W}_{1,3}{t}_{0}&-{z}_{2,3}^{2N}{W}_{2,3}{t}_{0}\end{array}\right],$$
(7)

where φq,α,b = Wq,αφq,α,a and \({W}_{q,\alpha }=E/({t}_{\alpha }^{-}+{z}_{q,\alpha }^{-1}{t}_{0})=({t}_{\alpha }^{+}+{z}_{q,\alpha }{t}_{0})/E\). To guarantee the nonzero value of φq,α,a, we have \(\det \left[M\right]=0\). Assuming that ∣z1,α∣ > ∣z2,α∣, so that the maximum ∣zμ∣ should be ∣z1,1z1,2z1,3∣ (μ = 1) and the minimum ∣zμ∣ should be ∣z2,1z2,2z2,3∣ (μ = 8). Hence, three cases can be listed. For g > 1, if ∣z1∣ > g we have the condition of valid solutions of ∣z1∣ = ∣z2∣. If ∣z1∣ = g, all solutions are valid. It is impossible that ∣z1∣ < g. For g = 1, if ∣z1∣ > 1 we have the condition of valid solutions of ∣z1∣ = ∣z2∣. If ∣z1∣ = 1, all solutions are valid. It is impossible that ∣z1∣ < 1. For g < 1, if ∣z1∣ > 1 we have the condition of valid solutions of ∣z1∣ = ∣z2∣. If ∣z1∣ = 1, all solutions are valid. It is impossible that ∣z1∣ < 1. All allowable eigenenergies and complete zq,α information can be numerically obtained by substituting these conditions into Eq. (4) under OBCs.

Once zq,α is determined, the propagation characteristics of fields are revealed. The restriction ∣zq,α∣ = 1 (both for q = 1, 2) indicates Bloch wave functions. The restriction ∣zq,α∣ > 1 (both for q = 1, 2) indicates significant accumulation in one direction, while the restriction ∣zq,α∣ < 1 (both for q = 1, 2) indicates significant accumulation in the opposite direction.

CROW simulations

Numerical simulations for CROW systems in this work are all performed using the 2D electromagnetic module of a commercial finite-element simulation software (COMSOL MULTIPHYSICS). In the eigenfrequency and eigenstate evaluations, the maximum element size for meshing is 45 nm (156 nm) in silicon (air) regions. All waveguide modes have an out-of-plane electric field polarization. The complete simulation region is a triangle with sidelength \(N+\frac{1}{4}\). So that enough air space can be created between edge/corner boundaries and simulation boundaries. Simulation boundaries are set as perfect electric conductors (PECs). Because of the excellent local performance of waveguide modes, this setup not only does not affect our numerical results but also reduces the computation time.