General construction of square-root lattices with arbitrarily degenerate edge states in double bulk gaps

Abstract

Square-root topological insulators can inherit edge states from an underlying topological parent system, but existing realizations are typically limited to low degeneracy and small winding numbers. Here, we introduce a general design strategy for extending square-root topological phases to an arbitrary winding number N by combining a fixed structural-subdivision framework with long-range couplings, enabling the number of inherited edge states to increase systematically without raising the root order. We reveal a parity-dependent winding-number phase diagram, where odd and even maximum winding numbers exhibit distinct phase-boundary structures and topological transition sequences. The resulting square-root lattices host N-fold degenerate topological edge states within each of two bulk band gaps, with the corresponding modes exhibiting characteristic fixed-position spatial localization. We also discuss experimentally accessible implementations in photonic waveguide arrays and topolectrical circuits. Our approach provides a scalable route to highly degenerate topological modes and multi-gap wave control in artificial lattices.

Graphical abstract

Keywords

double bulk gaps / edge states / winding number

1 Introduction

Topological insulators [1, 2] host edge states [3−5] protected by bulk topological invariants, providing a general framework for understanding robust phases of matter. In one dimension, the Su–Schrieffer–Heeger (SSH) model [6] is a paradigmatic example, where chiral symmetry protects a quantized winding number [7] that determines the existence of topological edge states. In its nearest-neighbor form, the winding number is restricted to w=0 or 1, which limits the number of edge modes. An effective route beyond this restriction is to introduce long-range couplings [8−17], which enrich the momentum-space winding of the Bloch Hamiltonian and enable topological phases with larger winding numbers [18, 19] and multiple edge states .

Among recent developments in artificial topological systems [20−28], square-root topology [29−31] has emerged as an unconventional mechanism for generating finite-energy edge states inherited from an underlying parent model. In contrast to ordinary topological lattices, a square-root Hamiltonian [32] is constructed such that its square reproduces a parent topological model in a block-diagonal form, thereby embedding the parent topology into an enlarged Hilbert space. This idea has been realized and extended in a variety of photonic [33−41], acoustic [42−44], and electrical platforms [45−49], and has stimulated further exploration of quartic-root [50, 51], 2ⁿ-root [52−57] topological phases. These studies have established square-root topology as a versatile route to engineering unconventional in-gap states and multi-gap spectral structures.

Despite this progress, existing approaches have largely focused on either increasing the root order to generate additional finite-energy gaps [52, 58] or increasing the winding number of the parent SSH model to obtain multiple zero-energy edge states. Although long-range hopping has recently been incorporated into square-root topology [59], a systematic route to independently control the edge-state multiplicity within each finite-energy gap at a fixed root order remains lacking. This raises a natural question: can a fixed square-root construction inherit an arbitrarily large parent winding number and thereby realize high-multiplicity edge states in each finite-energy gap?

In this work, we answer this question affirmatively. We realize N-fold degenerate topological edge states in a square-root extended SSH model by combining (2N−1)th-order long-range hopping in the parent chain with a fixed square-root structural-subdivision framework. The resulting descendant Hamiltonians inherit arbitrary parent winding numbers w = N, preserve chiral symmetry, and embed the high-winding parent topology into an enlarged Hilbert space without increasing the root order. We further reveal a parity-dependent evolution of the winding-number phase diagram, where odd and even maximum winding numbers follow distinct phase-boundary structures and transition sequences. To construct finite systems, we introduce an open-boundary completion scheme that retains all long-range hopping processes and the chiral bipartite structure, enabling a consistent realization of the N-fold edge states. The corresponding finite-chain spectra and eigenstate profiles further reveal characteristic fixed-position boundary localization determined by the internal structure. Finally, we discuss feasible implementations in photonic waveguide arrays and topolectrical circuits, suggesting a practical route toward multi-gap topological wave control in artificial lattice platforms.

2 Hamiltonian and topological characterization

We begin by considering an extended SSH model with long-range couplings. As shown in Fig. 1(a), the system consists of alternating sub-lattices A and B within each unit cell. The nearest-neighbor intracell and intercell hopping amplitudes are denoted by v_0 and v_1, respectively. To generate richer topological phases beyond the conventional SSH model, we further introduce higher-order long-range couplings v_N, which connect B_i to A_{i+N}. Here, N denotes the displacement in unit-cell index between the two coupled sites, corresponding to the (2N−1)th-nearest-neighbor hopping. In this generalized framework, the model incorporates hopping processes up to the (2N-1)th-order. Under periodic boundary conditions, the Bloch Hamiltonian can be written in momentum space as

(1)H_{ext}(k)=h_x(k){\sigma }_x+h_y(k){\sigma }_y,

where {\sigma }_{x,y} are the Pauli matrices acting on the sub-lattice space. The components h_x(k) and h_y(k) take the form

(2)h_x(k)={\displaystyle \sum _n{v}_n\cos (nk),h_y(k)={\displaystyle \sum _n{v}_n\sin (nk).}}

Equivalently, the Hamiltonian (1) can be expressed in the chiral off-diagonal representation

(3)H_{ext}(k)=\left(\begin{array}{cc}\begin{array}{c}0\\ h{(k)}^{\ast }\end{array}& \begin{array}{c}h(k)\\ 0\end{array}\end{array}\right),

with

(4)h(k)={\displaystyle \sum _n{v}_n{\mathrm{e}}^{-\mathrm{i}nk}.}

The summation provides a compact representation of all possible hopping processes. In the present work, only three hopping amplitudes are taken to be nonzero,

(5)v_n=\{\begin{array}{l}{v}_0,n=0,\\ v_1,n=1,\\ v_N,n=N,\\ 0,\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}.\end{array}

Thus,

(6)h(k)=v_0+v_1{\mathrm{e}}^{-\mathrm{i}k}+v_N{\mathrm{e}}^{-\mathrm{i}Nk}.

The integer N specifies the hopping range and enters solely through the phase factor {\mathrm{e}}^{-\mathrm{i}Nk}. The Hamiltonian (3) preserves chiral symmetry, satisfying

(7){\sigma }_z{H}_{ext}(k){\sigma }_z=-H_{ex}{}_t(k),

which ensures that the energy spectrum is symmetric with respect to zero energy. The inclusion of higher-order long-range couplings introduces additional harmonic components in h(k), thereby enriching the possible winding numbers and associated topological phases.

Within this square-root structural-subdivision framework, we insert auxiliary lattice sites into each hopping channel of the parent chiral lattice, thereby factorizing the original coupling into two-step hopping processes while preserving the bipartite structure. Concretely, auxiliary sites C, D and E are introduced between the original sub-lattices A and B, thereby enlarging the Hilbert space while maintaining a bipartite structure. The original hopping amplitudes v_0, v_1 and v_N are correspondingly replaced by their square-root amplitudes \sqrt{v_0}, \sqrt{v_1} and \sqrt{v_N}. The enlarged unit cell contains five sites, resulting in a fixed 5 × 5 Bloch Hamiltonian.

This construction implements the principle of “square-root after coupling”, meaning that the new Hamiltonian is designed so that its square reproduces the parent Hamiltonian in a block-diagonal manner. Under periodic boundary conditions, the Bloch Hamiltonian of the square-root extended SSH model can be written in the general form

(8)H_{root}(k)=\left(\begin{array}{cc}\begin{array}{c}{O}_{2\times 2}\\ Q{(k)}^{\ast }\end{array}& \begin{array}{c}Q(k)\\ O_{3\times 3}\end{array}\end{array}\right),

where

(9)Q(k)=\left(\begin{array}{ccc}\sqrt{v_0}& \sqrt{v_1}{\mathrm{e}}^{-\mathrm{i}k}& \sqrt{v_N}{\mathrm{e}}^{-\mathrm{i}Nk}\\ \sqrt{v_0}& \sqrt{v_1}& \sqrt{v_N}\end{array}\right).

The square-root Hamiltonian in Eq. (8) naturally inherits an exact chiral symmetry. Defining the chiral operator \mathrm{\Gamma }=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(I_2,-I_3) yields \left\{\mathrm{\Gamma },H_{root}(k)\right\}=0, here I_2 and I_3 represent the 2 × 2 and 3 × 3 identity matrices, respectively. The same symmetry is preserved for any winding number N because the off-diagonal structure remains unchanged regardless of the hopping range. Direct multiplication yields

(10)H_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}^2(k)=\left(\begin{array}{cc}\begin{array}{c}Q(k)Q{(k)}^{†}\\ 0\end{array}& \begin{array}{c}0\\ Q{(k)}^{†}Q(k)\end{array}\end{array}\right).

Equation (10) also reveals the standard algebraic factorization of supersymmetric quantum mechanics, where Q(k)Q{(k)}^{†} and Q{(k)}^{†}Q(k) constitute supersymmetric partner sectors with identical nonzero spectra, analogous to supersymmetric structures in Dirac and pseudo-Landau systems [60−62].

The upper block satisfies

(11)\begin{array}{rl}& Q(k)Q{(k)}^{†}=H_{ext}(k)+\mathrm{\Delta },\mathrm{\Delta }=v_0+v_1+v_N,\\ & H_{aux}(k)=Q{(k)}^{†}Q(k),\end{array}

showing that the parent Hamiltonian is recovered up to a constant energy shift. For finite lattices, the explicit forms of the finite-lattice Hamiltonian corresponding to Eq. (8) and H_{aux} are presented in Appendix A. The square-root construction therefore preserves the symmetry class and embeds the parent Hamiltonian into an enlarged Hilbert space without increasing the number of independent hopping orders.

We map the winding number,

(12)w={\displaystyle \frac{1}{2\text{π}\mathrm{i}}}{\displaystyle {\int }_{-\text{π}}^{\text{π}}\mathrm{d}k{\displaystyle \frac{\mathrm{d}}{\mathrm{d}k}}\ln h(k),h(k)=h_x(k)-\mathrm{i}{h}_y(k),}

in the parameter space spanned by v_1/v_0 and v_N/v_0. For generality, v_0=1. Since the highest-order term in Eq. (3) is (2N-1), larger N allows higher winding numbers and leads to a correspondingly more intricate phase diagram. Accordingly, v_N is not chosen arbitrarily, but is selected from the corresponding high-winding regions of Figs. 1(b, c) according to the desired winding number. Moreover, the phase diagrams for odd and even N exhibit qualitatively different structures, so we discuss the two cases separately. All topological regions are labeled explicitly. For odd N [Fig. 1(b)], the topological phase with the largest winding number occupies the deep orange region, while the w=1 and trivial w=0 phases are shown by light orange and white regions, respectively. By contrast, for even N [Fig. 1(c)], the highest-winding phase appears in the light blue region and carries an even winding number.

The structure of the phase-transition region also differs markedly between the two cases, the complete phase diagrams, their geometric origin, and the construction of arbitrary winding numbers are further discussed in Appendix B. For odd N, the phase diagram is asymmetric with respect to v_1=0. On the side, the winding number evolves from w=2 to w=N-1 in steps of two. whereas on the v_1>0 side, it runs from w=3 to w=N-2. For even N, the phase diagram is symmetric about v_1=0, and the winding number increases from w=2 to w=N-2 in steps of two on both sides. To further clarify this evolution, we fix v_N=3 and plot the winding number as a function of v_1. For odd N=3,5,7 [Fig. 1(d)], the phase transitions are more densely distributed on the side, consistent with the asymmetry of the phase diagram. By contrast, for even N=2,4,6 [Fig. 1(e)], the winding number evolution is mirror-symmetric with respect to v_1.

3 Bulk spectra under periodic boundary conditions

To illustrate the effect of increasing long-range couplings on the bulk topology, we calculate the bulk spectra of the extended SSH model and its square-root counterpart. As shown in Fig. 2(a), the bulk spectrum of the extended SSH model undergoes a systematic evolution as the highest hopping order from 5 to larger values. For fixed parameters v_0=1, v_1=1 and v_N=5, the winding number increases sequentially as 3, 4, 5,···. This behavior originates from the growing contribution of higher-order hopping processes, which generate a more intricate momentum dependence in the Bloch Hamiltonian and drive successive band inversions in momentum space.

Figure 2(b) displays the corresponding bulk spectra of the square-root extended SSH model for the same set of parameters, all taken within the topological phase. The square-root construction enlarges the Hilbert space and reconstructs the bulk spectrum, producing additional bands and gaps associated with the auxiliary degrees of freedom. Nevertheless, the essential topological features inherited from the parent model remain unchanged. In particular, the increasingly structured bulk dispersions generated by higher-order hopping in the parent model are preserved after the square-root operation. The topological evolution driven by increasing hopping order is therefore retained in the square-root model. This comparison also clarifies the distinct roles of the parent and square-root systems. The parity-dependent topological evolution originates from the parent extended SSH Hamiltonian, whereas the square-root construction reorganizes the inherited high-winding topology into the multigap spectrum of the enlarged descendant lattice. The corresponding bulk dispersions at the phase-transition points are provided in Appendix C.

4 Edge states under open boundary conditions

4.1 Finite-chain construction and open-boundary completion

To investigate the topological edge states, a finite lattice realization is required. For long-range hopping, naive truncation generates incomplete boundary couplings and consequently breaks the chiral bipartite structure of the Hamiltonian. We therefore employ a boundary-completion scheme that retains all hopping terms under open boundary conditions. The construction for the representative case N=2 is illustrated in Fig. 3(a). We consider a finite chain composed of L unit cells, where i=1,\cdots ,L labels the unit cell index. Each unit cell contains five lattice sites: the primary sites A_i and B_i, together with the auxiliary sites C_i, D_i, and E_i generated by the square-root construction. Specifically, the three subdivided coupling channels are A_i−C_i−B_i, B_i−D_i−A_{i+1}, and B_i−E_i−A_{i+N}, corresponding to the parent hoppings v_0, v_1, and v_N, respectively. Thus, E_i is an auxiliary site inserted into the long-range B_i−A_{i+N} coupling channel.

Because the hopping amplitude v_2 connects sites separated by two unit cells, the boundary sites B_{L-1} and B_L require additional A-type sites to receive the corresponding hopping processes. To retain all couplings under open boundary conditions, two auxiliary sites A_{L+1} and A_{L+2} are introduced at the right boundary. Following the square-root construction rule, the corresponding auxiliary sites E_{L-1} and E_L are subsequently inserted, ensuring that the number of added E-type sites matches that of the boundary A-type sites. This construction can be generalized to arbitrary odd-order hopping n=2N-1. The coupling structure is summarized in Fig. 3(b), which shows representative cases for n=3, 5 and 7. Accordingly, an order-n hopping process requires N auxiliary A-type sites at the boundary to preserve all hopping terms. For a finite chain containing L unit cells with highest hopping order 2N-1, the total number of lattice sites is

(13)N_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=L(N_{AB}+N_C+N_{br}),

where N_{AB} corresponds to the primary lattice sites A and B, N_C denotes the intracell auxiliary site C, and N_{br} counts the auxiliary sites introduced by intercell hopping branches. In the present model N_{br}=2, corresponding to the nearest- and next-nearest-neighbor auxiliary sites D and E. This counting also clarifies how the boundary completed square-root lattice provides a consistent finite-size framework for realizing high-winding square-root phases and their associated edge states.

4.2 Finite-chain spectra and N-fold degenerate edge states

To investigate the boundary properties of the square-root extended SSH model, we calculate the energy spectra of finite chains under open boundary conditions. The spectra as a function of the coupling ratio v_1/v_0 are shown in Fig. 4 for several representative long-range hopping orders, with the other parameters fixed. As the highest hopping order increases, the number of in-gap edge state branches grows systematically, reflecting the increasing winding number inherited from the parent Hamiltonian. For the 5th-nearest-neighbor case [Fig. 4(a)], each of the two bulk gaps hosts a threefold degenerate set of edge states. For the 7th-nearest-neighbor case [Fig. 4(b)], each of the two bulk gaps hosts a fourfold degenerate set of edge states. For still higher hopping order, exemplified by the 13th-nearest-neighbor case [Fig. 4(c)], each gap hosts a sevenfold degenerate set of edge states, providing the spectral signature of the higher-winding square-root phase. To make the edge-state degeneracies more explicit, Figs. 4(d−f) show magnified views of the upper-gap spectra at representative coupling ratios, resolving the threefold, fourfold, and sevenfold edge-state multiplicities for w = 3, w = 4, and w = 7, respectively. As the coupling ratio v_1/v_0 varies, the edge state branches remain pinned inside the corresponding bulk gaps, consistent with the bulk-edge correspondence.

To further characterize the localization properties of these states, we calculate the inverse participation ratio (IPR):

(14)IPR=\frac{{{\displaystyle \sum }}_i{\left|{\psi }_i\right|}^4}{{\left({{\displaystyle \sum }}_i{\left|{\psi }_i\right|}^2\right)}^2},

where {\psi }_i denotes the amplitude of the eigenstate at lattice site i. A larger IPR indicates stronger spatial localization. The IPR distribution clearly distinguishes the in-gap edge modes from the extended bulk states. Moreover, as the hopping order increases, the number of highly localized in-gap states grows systematically, consistent with the increasing edge state degeneracy in each bulk gap. The corresponding real-space profiles of representative edge modes are shown below.

The eigenvalue spectra with IPR encoded by color are shown in Fig. 5(a) for the w=3 phase and in Fig. 5(c) for the w=4 phase. In both spectra, the majority of states have small IPR values and correspond to extended bulk modes, whereas the in-gap states exhibit significantly enhanced IPR and are therefore identified as edge modes. Notably, an additional high-IPR state appears within the bulk spectrum near mode number ~55. Its localized character is distinct from the winding-number-induced edge-state multiplets and is associated with a BIC-like interference-induced localization mechanism [28, 39, 63]. Four representative eigenstates, labeled I−IV, are selected from each spectrum for further analysis, and their spatial profiles are shown in Figs. 5(b) and (d).

For the w=3 phase [Fig. 5(a)], one selected state is extended across the lattice, whereas the other three are localized near the boundary. A similar distinction is found for the w=4 phase [Fig. 5(c)], where the selected in-gap states likewise exhibit clear boundary localization. The parent zero-energy edge modes are correspondingly mapped to the finite energies E=\pm \sqrt{\mathrm{\Delta }} in the square-root spectrum. In contrast to the parent extended SSH model (see Appendix D), whose edge modes appear symmetrically at both ends, the square-root descendant exhibits single-sided localization. This feature arises from the enlarged five-site unit cell together with the asymmetric termination introduced by the open-boundary completion scheme. The dependence of the edge-state localization on the boundary termination is further demonstrated in Appendix E, where alternative boundary completions lead to corresponding changes in the localization side. The edge modes predominantly occupy the primary lattice sites A and B. For a finite chain with L=20 unit cells, corresponding to 100 sites, the three nearly degenerate edge states are localized near the right boundary, with dominant peaks at sites 97, 92, and 87. More generally, for a winding number w, the corresponding boundary modes extend over the last w enlarged unit cells. The fixed spacing between these peaks is determined by the internal structure of the square-root unit cell, providing a real-space signature of the N-fold degenerate edge state distribution enabled by the arbitrary-winding square-root construction.

Beyond increasing the number of boundary states, a high winding number provides additional topological degrees of freedom through the distinct spatial distributions of the boundary modes, offering possibilities for mode-selective excitation and multi-channel routing or splitting [64, 65]. Their multiplicity remains robust against moderate coupling disorder, as demonstrated in Appendix F, supporting the potential for robust topological transport.

4.3 Experimental feasibility

The present square-root extended SSH model can be implemented in artificial platforms such as photonic waveguide arrays and topolectrical circuits. For photonic waveguide arrays, the continuous paraxial description can be reduced to an effective tight-binding model in the single-mode and weak-coupling regime, as detailed in Appendix G. In this scheme, photonic lattices can be fabricated using femtosecond laser direct writing, which enables precise control of the geometric arrangement of waveguides. Following the strategy demonstrated in Ref. [12], multilayer waveguide architectures can be designed to realize effective long-range couplings v_N. In this platform, the square-root lattice is implemented by inserting additional waveguides that correspond to the auxiliary sites introduced in the square-root construction [38]. The effective hopping amplitudes are determined by the spatial separations between waveguides, which decay approximately exponentially with distance. By carefully engineering the inter-waveguide distances, the desired coupling configuration of the square-root Hamiltonian (8) can be faithfully reproduced.

Alternatively, the tight-binding Hamiltonian can be emulated by a network of capacitors and inductors on a printed circuit board [66−68]. Each lattice site (A,B,\cdots ,E) corresponds to a circuit node connected to ground via an inductor (L_a,L_b,\cdots ,L_e) and to neighboring nodes via coupling capacitors (C_0,C_1,C_N). The hopping amplitudes are encoded in the capacitances. By tuning the driving frequency to the resonance condition at which the diagonal onsite terms vanish, the admittance spectrum obtained from standard impedance measurements reveals the band structure and the emergence of in-gap topological edge states.

5 Conclusions

In summary, we have established a fixed-order square-root construction that systematically inherits arbitrary winding numbers from long-range SSH parent chains. Starting from extended parent Hamiltonians with dominant (2N-1)th-order hopping, we obtain chiral square-root descendants with a fixed five-site unit cell. These higher-winding square-root phases exhibit two spectrally separated finite-energy gaps, each hosting an N-fold degenerate edge state. We further introduce an open-boundary completion scheme that preserves the long-range hopping structure and the chiral bipartite character in finite chains. Numerical spectra, inverse-participation-ratio calculations, and real-space eigenstate profiles confirm the systematic increase of edge state degeneracy with parent winding number and reveal a characteristic single-sided localization in the square-root lattice. This identifies parent-winding engineering as a systematic route to highly degenerate square-root phases in long-range chiral lattices.

Furthermore, long-range coupling has also been used to enhance boundary-mode multiplicity in higher-order topological systems [13, 69], including coupling-inverted acoustic lattices with long-range coupling exceeding nearest-neighbor coupling. This suggests a possible extension of the present square-root framework to higher-order parent lattices, where high-multiplicity corner or hinge modes could be mapped into finite-energy gaps, provided that the relevant symmetries and long-range coupling hierarchy are preserved.

6 Appendix A : Hamiltonian in a finite structure and the specific form of H_res

The Bloch Hamiltonian in momentum space is given by [see Eq. (8) of the main text]. Under periodic boundary conditions, we define the Bloch basis

(A1)\begin{array}{c}{C}_k=(c_{k,A},c_{k,B},c_{k,C},c_{k,D},c_{k,E}).\end{array}

To obtain the real-space representation, we perform the inverse Fourier transform on the Bloch Hamiltonian

(A2)\begin{array}{c}{c}_{j,\alpha }={\displaystyle \frac{1}{\sqrt{L}}}{\displaystyle \sum _{j=1}^L{\mathrm{e}}^{\mathrm{i}kj}{c}_{k,\alpha },}\end{array}

and substitute into H={\sum }_k{C}_k^{†}{H}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}(k)C_k. After collecting terms, the real-space Hamiltonian is obtained as

(A3)\begin{array}{c}\begin{array}{rl}H=& \sum _{j=1}^L[\sqrt{v_0}(c_{j,A}^{†}{c}_{j,C}+c_{j,B}^{†}{c}_{j,C}+h.c.)\\ & +\sqrt{v_1}(c_{j+1,A}^{†}{c}_{j,D}+c_{j,B}^{†}{c}_{j,D}+h.c.)\\ & +\sqrt{v_N}(c_{j,B}^{†}{c}_{j,E}+c_{j+N,A}^{†}{c}_{j,E}+h.c.)],\end{array}\end{array}

where c_{j,\alpha } denotes the annihilation operator for site \alpha in the j-th unit cell, and L is the number of unit cells. We now write the Hamiltonian matrix in the basis ordered as

(A4)\begin{array}{c}\begin{array}{rl}& (A_1,B_1,C_1,D_1,E_1,A_2,B_2,C_2,D_2,E_2,\cdots ,\\ & A_L,B_L,C_L,D_L,E_L{)}^{\mathrm{T}}.\end{array}\end{array}

The resulting 5L\times 5L matrix is block-sparse, we denote each 5 × 5 block by its row and column unit-cell indices. The intra-cell block (i=j) is

(A5)\begin{array}{c}{H}_{i,i}=\left(\begin{array}{ccccc}0& 0& \sqrt{v_0}& 0& 0\\ 0& 0& \sqrt{v_0}& \sqrt{v_1}& \sqrt{v_N}\\ \sqrt{v_0}& \sqrt{v_0}& 0& 0& 0\\ 0& \sqrt{v_1}& 0& 0& 0\\ 0& \sqrt{v_N}& 0& 0& 0\end{array}\right).\end{array}

The nearest-neighbor inter-cell block from cell i to cell i+1 is

(A6)\begin{array}{c}{H}_{i+1,i}=\left(\begin{array}{ccccc}0& 0& 0& \sqrt{v_1}& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right),H_{i,i+1}=H_{i+1,i}^{\mathrm{T}}.\end{array}

The N-th order inter-cell block from cell i to cell i+N is

(A7)\begin{array}{c}{H}_{i+N,i}=\left(\begin{array}{ccccc}0& 0& 0& 0& \sqrt{v_N}\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right),H_{i,i+N}=H_{i+N,i}^{\mathrm{T}}.\end{array}

The full Hamiltonian thus takes the block-banded form

(A8)\begin{array}{c}H=\left(\begin{array}{ccccccc}{H}_{1,1}& H_{1,2}& \cdots & H_{1,1+N}& 0& 0& 0\\ H_{2,1}& H_{2,2}& H_{2,3}& \cdots & H_{2,2+N}& 0& 0\\ ⋮& H_{3,2}& H_{3,3}& \ddots & 0& \ddots & 0\\ H_{1+N,1}& ⋮& \ddots & \ddots & ⋮& 0& H_{L-N,L}\\ 0& H_{2+N,2}& 0& \cdots & H_{L-2,L-2}& H_{L-2,L-1}& ⋮\\ 0& 0& \ddots & 0& H_{L-1,L-2}& H_{L-1,L-1}& H_{L-1,L}\\ 0& 0& 0& H_{L,L-N}& 0& H_{L,L-1}& H_{L,L}\end{array}\right),\end{array}

which is enlarged 5\times 5 blocks. This matrix is used for numerical diagonalization to obtain the energy spectra and eigenstate profiles discussed in the main text.

While the finite-size construction above is suitable for open-boundary calculations, the topological properties of the model are most conveniently analyzed in momentum space under periodic boundary conditions. In this case, the square of the Hamiltonian assumes a block-diagonal, and the residual 3\times 3 block H_{aux}, which captures the effective couplings among the auxiliary degrees of freedom, takes the explicit form

(A9)\begin{array}{c}{H}_{aux}=\left(\begin{array}{ccc}2v_0& \sqrt{v_0{v}_1}\left({\mathrm{e}}^{-\mathrm{i}k}+1\right)& \sqrt{v_0{v}_N}\left({\mathrm{e}}^{-\mathrm{i}Nk}+1\right)\\ \sqrt{v_0{v}_1}\left({\mathrm{e}}^{\mathrm{i}k}+1\right)& 2v_1& \sqrt{v_1{v}_N}\left({\mathrm{e}}^{-\mathrm{i}(1-N)k}+1\right)\\ \sqrt{v_0{v}_N}\left({\mathrm{e}}^{\mathrm{i}Nk}+1\right)& \sqrt{v_1{v}_N}\left({\mathrm{e}}^{\mathrm{i}(1-N)k}+1\right)& 2v_N\end{array}\right).\end{array}

7 Appendix B : Winding-number phase diagrams and geometric characterization

We present in this appendix the numerically computed winding number phase diagrams for the extended SSH model with long-range couplings, which complement the schematic illustrations in [Figs. 1(b) and (c) of the main text]. These numerical results demonstrate that the winding number can be systematically increased by making the highest-order long-range coupling dominant. As seen in the phase diagrams (Fig. B1), the winding number is always of the same parity as N in the intermediate regions. This parity-dependent behavior is consistent with the analysis in Section 2 of the main text.

To further clarify the origin of the parity-dependent phase diagrams, we consider Eq. (6) of the main text. Under the momentum shift (k\to k+\text{π}),

(B1)\begin{array}{c}h(k+\text{π},v_1,v_N)=v_0-v_1{\mathrm{e}}^{-\mathrm{i}k}+{(-1)}^N{v}_N{\mathrm{e}}^{-\mathrm{i}Nk}.\end{array}

For even N, this transformation is equivalent to (v_1\to -v_1) at fixed v_N, so the two trajectories are identical up to a momentum shift and have the same winding number. For odd N, at fixed v_N, the trajectories for v_1 and -v_1 are inequivalent, resulting in the asymmetric phase diagram. Equivalently, with (z={\mathrm{e}}^{-\mathrm{i}k}), the polynomial [p_N(z)=v_0+v_{1}z+v_N{z}^N] exhibits the same parity-dependent transformation under (z\to -z). Representative winding trajectories are shown in Fig. B2.

The same polynomial representation also provides a constructive criterion for realizing the maximum winding number. According to Rouché’s theorem, when the

(B2)\begin{array}{c}\left|v_N\right|>\left|v_0\right|+\left|v_1\right|,\end{array}

polynomia p_N(z) has the same number of zeros inside the unit circle as the dominant term v_N{z}^N, namely N. Therefore, under the convention adopted here, the winding number is w=N. Since N is determined by the long-range hopping order, this condition provides a constructive route to arbitrarily large integer winding numbers.

Figure B3 provides a direct geometric visualization for N = 1,···, 6. In each case, the complex trajectory of h(k) encircles the origin exactly N times during one Brillouin-zone cycle, confirming the relation w=N.

8 Appendix C : Critical bulk dispersions of the square-root model

For representative winding numbers, the bulk band dispersions at the topological transition points are shown in Fig. C1. For the even-winding cases [Figs. C1(a) and (b)], the bulk gap closes under the condition v_1-v_0=v_N. For the odd-winding cases [Figs. C1(c) and (d)], the critical conditions become v_N+v_1=v_0, respectively. Owing to the square-root construction, the spectra are reorganized in the enlarged Hilbert space but remain gapless at the critical point. As the long-range hopping order increases, the bulk dispersions develop richer fine structures, indicating that higher-order couplings play an increasingly important role. This behavior provides a spectral signature of the criticality inherited from the parent high-winding phases and reveals the spectral origin of the topological phase transitions in the extended model.

9 Appendix D : Edge state localization of arbitrary-order extended SSH model

For the same parameters as those used in Fig. 5 of the main text, Fig. D1 presents the eigenvalue spectra and representative eigenstates of the parent extended SSH model. The in-gap states in Figs. D1(a) and (c) are characterized by enhanced IPR, and their spatial profiles in Figs. D1(b) and (d) confirm boundary localization, in contrast to the extended bulk states. These results clarify the parent-state origin of the edge modes and their inherited spectral and localization characteristics in the square-root mode.

10 Appendix E : Effect of boundary termination on the localization of the topological edge states

we further examined different boundary terminations and recalculated both the energy spectra and spatial distributions of the in-gap states, with representative results for w = 3 shown in Fig. E1. When the same boundary completion is introduced at both ends [Figs. E1(a) and (b)], the finite lattice becomes symmetrically terminated, and the edge states are localized at both ends, resulting in a total of 2N-fold boundary states in each topological gap. When the same additional boundary sites are introduced only at the left end [Figs. E1(c) and (d)], the boundary states are correspondingly transferred to the left boundary.

11 Appendix F: Robustness of N-fold end states

To examine the robustness of the N-fold edge states, the relevant couplings are randomized within ±12.5%. Although the in-gap edge-state energies fluctuate under disorder, their multiplicities remain fixed at 3, 4, and 7, confirming the robustness of the boundary-state manifold.

12 Appendix G: Connection between the continuous and tight-binding models

For a photonic waveguide array, light propagation under the paraxial approximation can be described by

(G1)\begin{array}{c}\mathrm{i}{\displaystyle \frac{\mathrm{\partial }\mathrm{\Psi }(x,y,z)}{\mathrm{\partial }z}}=H_{\mathrm{\perp }}\mathrm{\Psi }(x,y,z),\end{array}

where H_{\mathrm{\perp }} describes the transverse diffraction and refractive-index distribution of the waveguide array. Projecting Eq. (G1) onto the localized fundamental modes yields

(G2)\begin{array}{c}\mathrm{i}{\displaystyle \frac{\mathrm{d}{a}_j}{\mathrm{d}z}}={\beta }_j{a}_j+{\displaystyle \sum _{l\ne j}{C}_{jl}{a}_l,}\end{array}

where C_{jl} corresponds to the hopping amplitude in the tight-binding Hamiltonian. Thus, the tight-binding model provides an effective description of the continuous photonic system. In the multilayer geometry shown in Fig. G1, waveguides that are distant in the effective one-dimensional lattice can be brought into closer physical proximity, providing an experimentally demonstrated route to long-range couplings that can exceed the nearest-neighbor coupling.

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