Long-range chiral pairing enables topological superconductivity in triangular lattices without spin−orbit coupling and magnetic field

Abstract

This paper demonstrates a pathway to topological superconductivity in monolayer triangular lattices through long-range pairing without requiring spin−orbit coupling and magnetic field, contrasting conventional frameworks reliant on superconductivity and spin−orbit coupling and time-reversal symmetry (TRS) breaking. Berry curvature analysis reveals spontaneous TRS-breaking-induced peaks or valleys under long-range pairing, signaling nontrivial topology superconducting states. Notably, the increase in the long-range pairing strength only changes the size of the energy band-gap, without triggering a topological phase transition. This characteristic is verified by calculating Berry curvature and topological edge states. In zigzag and armchair-edge ribbons of finite width, the topological edge states are influenced by the ribbon boundary symmetry and the interaction range of long-range pairing. Under nearest-neighbor pairing, the topological edge states maintain particle−hole symmetry and match the corresponding Chern number. However, next-nearest-neighbor and third-nearest-neighbor pairings break the particle-hole symmetry of the topological edge states in armchair-edge ribbon. This work proposes a mechanism to realize topological superconductivity without relying on spin−orbit coupling and magnetic field, thereby offering a theoretical foundation for simplifying the design of topological quantum devices.

Graphical abstract

Keywords

long-range chiral pairing / topological superconducting state / spin−orbit coupling / magnetic field

1 Introduction

Topological superconductors are characterized by a bulk band gap induced by superconducting pairing potentials and topological edge states that host Majorana fermions with non-Abelian statistics, which hold great promise for topological quantum computing [1−3]. The most straightforward way to implement topological superconductors is to search for odd parity p-wave superconductors, which are extremely rare in nature. Researchers have subsequently turned to the study of artificial topological superconductors and proposed several schemes to achieve topological superconductivity [4], such as the proximity effect [5], chemical doping [6−8], and external field modulation [9]. However, these external control methods generally require precise control of external conditions and may lead to unstable outcomes. And previous study also shows local (say, electric or magnetic) fields do not manipulate the quantum information [10].

Theoretical research indicates that Majorana zero mode-dependent odd−parity pairing (such as p-wave pairing) often involves electron interactions between different lattice sites (i.e., long-range pairing) [11]. Recent reports have shown that, under long-range pairing superconducting states, time-reversal symmetry (TRS) can be spontaneously broken, inducing a topological superconducting (TSC) state under zero magnetic field, which significantly reduces the experimental requirements [12−17]. Of particular note is that long-range interactions can not only serve as an alternative path to topological superconductivity but can also actively enhance the topological order. As demonstrated by Viyuela et al. [18] in a two-dimensional p-wave superconductor with long-range hopping and pairing amplitudes, long-range couplings can significantly enlarge the regime of a topological chiral phase in the parameter space. This “enhancement” effect dramatically reduces the experimental demand for precise tuning of the chemical potential, providing a broader platform for realizing and observing topological superconducting states. Monolayer MoS₂, considering the nearest-neighbor (NN) spin-singlet pairing potential, TSC phases with non-zero Chern number (CN) and spontaneous TRS breaking is realized [14]. The graphene model with NN pairing presents TSC state with CNs of 1 and 2 under zero magnetic field [15]. The two-dimensional D_4h point symmetric square lattice achieves a mixed singlet superconducting pairing function that combines on-site and long-range pairing, exhibiting topologically nontrivial high CNs and Majorana zero modes located outside high symmetry points [16]. The checkerboard model with next-nearest-neighbor (NNN) pairing displays multiple TSC states with CN as high as 4 when the net magnetic field is zero [17]. However, in certain cases, long-range pairing superconductors have numerous non-topological energy bands near the Fermi level (E_F). An excess of non-topological energy bands can easily obscure the topological electronic behavior. Moreover, topological edge states with some CNs are not always robust, and the relationship between CNs and topological edge states does not always adhere to the conventional bulk−boundary correspondence principle [1]. The underlying physical mechanism behind this mismatch between CN and topological edge state count remains unclear and requires comprehensive and in-depth research.

In this paper, we derive long-range pairing functions (including NN, NNN, and third-nearest-neighbor (TNN) pairings) with higher-dimensional irreducible lattice representations E₂ for the triangular lattice possessing C_6v point group symmetry by using the projection operator method [19]. The singlet long-range pairing function corresponds to d-wave symmetry and naturally suggests possible chiral superconductivity phases [20]. Potential applications of anisotropic d + id' superconducting pairings have been proposed in materials such as water-intercalated sodium cobaltates, bilayer silicene, the epitaxial bilayer films of bismuth and nickel [21−23]. To facilitate the experimental regulation, this paper investigates systematically TSC state by considering chiral d+id' pairing under zero magnetic field based on a Bogoliubov-de Gennes (BdG) Hamiltonian. We initially employ efficient method [24] to study CN as a function of chemical potential and long-range pairing strength. Subsequently, we calculate the topological edge states of zigzag and armchair ribbons to confirm TSC state of CNs phase diagram. To comprehensive study of the topological edge state of the system, we further investigated the probability distributions |ψ(n)|² near the E_F.

2 Model and method

The model under consideration is a triangular lattice with the C_6v point group. The paper primarily interested in studying the chiral superconducting states d+id', which can be described by pairing on NN and NNN and TNN neighbors. These pairings are initially incorporated into our model, so the tight-binding Hamiltonian defined on a triangular lattice becomes [25]

(1)H=H_t+H_{sc},

(2)H_t=-{\displaystyle \sum _{i,j,\sigma }{t}_{ij}{c}_{i\sigma }^{†}{c}_{j\sigma }-\mu {\displaystyle \sum _{i,\sigma }{c}_{i\sigma }^{†}{c}_{i\sigma },}}

(3)H_{\mathrm{S}\mathrm{C}}={\displaystyle \sum _{⟨i,j⟩}{\mathrm{\Delta }}_{ij}\left(c_{i\uparrow }^{†}{c}_{j\downarrow }^{†}+c_{i\downarrow }{c}_{j\uparrow }\right),}

where the hopping term H_t contains the hopping amplitudes t_ij and the chemical potential μ. The last term describes long-range pairing, and the ⟨i,j⟩ describe the i and j neighboring pairings (including NN, NNN or TNN).

Previous study introduces long-range pairing potential based on the projection operator approach [19]. The trial wave functions of three types of neighbors [NN, NNN, TNN in Fig. 1(a)] are

(4)\begin{array}{rl}& {\varphi }_1^{\mathrm{N}\mathrm{N}}={\delta }_{i,i+x},{\varphi }_2^{\mathrm{N}\mathrm{N}}={\delta }_{i,\left(i+\frac{x}{2},i+\frac{\sqrt{3}y}{2}\right)},\\ & {\varphi }_1^{\mathrm{N}\mathrm{N}\mathrm{N}}={\delta }_{i,i+\sqrt{3}y},{\varphi }_2^{\mathrm{N}\mathrm{N}\mathrm{N}}={\delta }_{i,\left(i-\frac{3x}{2},i+\frac{\sqrt{3}y}{2}\right)},\\ & {\varphi }_1^{\mathrm{T}\mathrm{N}\mathrm{N}}={\delta }_{i,i+2x},{\varphi }_2^{\mathrm{T}\mathrm{N}\mathrm{N}}={\delta }_{i,\left(i+x,i+\sqrt{3}y\right)}.\end{array}

We make use of a fundamental projection theorem stating that the operator p(E_2)=\sum _g{\chi }_i^{\ast }(g)g projects out the contribution which transforms in the irreducible representation E₂. Here, the sum runs over all point-group operations g with the corresponding complex-conjugate characters {\chi }_i^{\ast }(g). we then apply the projection operator p(E₂) to trial wave functions Eq. (4) and obtain the following NN and NNN and TNN basis functions for the trivial representation E₂:

(5)\begin{array}{rl}p\left(E_2\right){\varphi }_1^{\mathrm{N}\mathrm{N}}=& {\chi }_i^{\ast }\left(E\right){\delta }_{i,i+x}+{\chi }_i^{\ast }\left(C_2\right){\delta }_{i,i-x}+{\chi }_i^{\ast }\left(C_3\right){\delta }_{i,\left(i-\frac{x}{2},i+\frac{\sqrt{3}y}{2}\right)}\\ & +{\chi }_i^{\ast }\left(C_3^{-1}\right){\delta }_{i,\left(i-\frac{x}{2},i-\frac{\sqrt{3}y}{2}\right)}+{\chi }_i^{\ast }\left(C_6\right){\delta }_{i,\left(i+\frac{x}{2},i+\frac{\sqrt{3}y}{2}\right)}+{\chi }_i^{\ast }\left(C_6^{-1}\right){\delta }_{i,\left(i+\frac{x}{2},i-\frac{\sqrt{3}y}{2}\right)},\\ p\left(E_2\right){\varphi }_2^{\mathrm{N}\mathrm{N}}=& {\chi }_i^{\ast }\left(E\right){\delta }_{i,\left(i+\frac{x}{2},i+\frac{\sqrt{3}y}{2}\right)}+{\chi }_i^{\ast }\left(C_2\right){\delta }_{i,\left(i-\frac{x}{2},i-\frac{\sqrt{3}y}{2}\right)}+{\chi }_i^{\ast }\left(C_3\right){\delta }_{i,i-x}\\ & +{\chi }_i^{\ast }\left(C_3^{-1}\right){\delta }_{i,\left(i+\frac{x}{2},i-\frac{\sqrt{3}y}{2}\right)}+{\chi }_i^{\ast }\left(C_6\right){\delta }_{i,\left(i-\frac{x}{2},i+\frac{\sqrt{3}y}{2}\right)}+{\chi }_i^{\ast }\left(C_6^{-1}\right){\delta }_{i,i+x},\end{array}

(6)\begin{array}{rl}p\left(E_2\right){\varphi }_1^{\mathrm{N}\mathrm{N}\mathrm{N}}=& {\chi }_i^{\ast }\left(E\right){\delta }_{i,i+\sqrt{3}y}+{\chi }_i^{\ast }\left(C_2\right){\delta }_{i,i-\sqrt{3}y}+{\chi }_i^{\ast }\left(C_3\right){\delta }_{i,\left(i-\frac{3}{2}x,i-\frac{\sqrt{3}}{2}y\right)}\\ & +{\chi }_i^{\ast }\left(C_3^{-1}\right){\delta }_{i,\left(i+\frac{3}{2}x,i-\frac{\sqrt{3}}{2}y\right)}+{\chi }_i^{\ast }\left(C_6\right){\delta }_{i,\left(i-\frac{3}{2}x,i+\frac{\sqrt{3}}{2}y\right)}+{\chi }_i^{\ast }\left(C_6^{-1}\right){\delta }_{i,\left(i+\frac{3}{2}x,i-\frac{\sqrt{3}}{2}y\right)},\\ p\left(E_2\right){\varphi }_2^{\mathrm{N}\mathrm{N}\mathrm{N}}=& {\chi }_i^{\ast }\left(E\right){\delta }_{i,\left(i+\frac{3}{2}x,i+\frac{\sqrt{3}}{2}y\right)}+{\chi }_i^{\ast }\left(C_2\right){\delta }_{i,\left(i-\frac{3}{2}x,i-\frac{\sqrt{3}}{2}y\right)i-\sqrt{3}y}+{\chi }_i^{\ast }\left(C_3\right){\delta }_{i,\left(i-\frac{3}{2}x,i+\frac{\sqrt{3}}{2}y\right)}\\ & +{\chi }_i^{\ast }\left(C_3^{-1}\right){\delta }_{i,i-\sqrt{3}y}+{\chi }_i^{\ast }\left(C_6\right){\delta }_{i,i+\sqrt{3}y}+{\chi }_i^{\ast }\left(C_6^{-1}\right){\delta }_{i,\left(i+\frac{3}{2}x,i-\frac{\sqrt{3}}{2}y\right)},\end{array}

(7)\begin{array}{rl}p\left(E_2\right){\varphi }_1^{\mathrm{T}\mathrm{N}\mathrm{N}}=& {\chi }_i^{\ast }\left(E\right){\delta }_{i,i+2x}+{\chi }_i^{\ast }\left(C_2\right){\delta }_{i,i-2x}+{\chi }_i^{\ast }\left(C_3\right){\delta }_{i,\left(i-x,i+\sqrt{3}y\right)}\\ & +{\chi }_i^{\ast }\left(C_3^{-1}\right){\delta }_{i,\left(i-x,i-\sqrt{3}y\right)}+{\chi }_i^{\ast }\left(C_6\right){\delta }_{i,\left(i+x,i+\sqrt{3}y\right)}+{\chi }_i^{\ast }\left(C_6^{-1}\right){\delta }_{i,\left(i+x,i-\sqrt{3}y\right)},\\ p\left(E_2\right){\varphi }_2^{\mathrm{T}\mathrm{N}\mathrm{N}}=& {\chi }_i^{\ast }\left(E\right){\delta }_{i,\left(i+x,i+\sqrt{3}y\right)}+{\chi }_i^{\ast }\left(C_2\right){\delta }_{i,\left(i-x,i-\sqrt{3}y\right)}+{\chi }_i^{\ast }\left(C_3\right){\delta }_{i,i-2x}\\ & +{\chi }_i^{\ast }\left(C_3^{-1}\right){\delta }_{i,\left(i+x,i-\sqrt{3}y\right)}+{\chi }_i^{\ast }\left(C_6\right){\delta }_{i,\left(i-x,i+\sqrt{3}y\right)}+{\chi }_i^{\ast }\left(C_6^{-1}\right){\delta }_{i,i+2x}.\end{array}

By transforming Eqs. (5)−(7) into k-space, we obtain the following long-range pairing functions (∆_E2NN, ∆_E2NNN, ∆_E2TNN) Eqs. (8)−(11) for the two-dimensional representation E₂ with strength parameters. In real materials, pairing channels at different distances often coexist and interact. For instance, in the honeycomb lattice, NN and NNN d+id'-wave pairings can not only coexist but also mutually enhance each other [26]. Similarly, in the triangular lattice, the NN, NNN, and TNN pairings all fall within the same E₂ irreducible representation, allowing them to mix naturally. To systematically model their interplay, we employ a spherical coordinate parameterization in which the relative strength of the long-range pairing is governed by the azimuthal angle θ and the polar angle φ, as illustrated in Fig. 1(b),

(8)\begin{array}{rl}{\mathrm{\Delta }}_{E2\mathrm{N}\mathrm{N}}^{d_{x^2-y^2}}(\mathit{k})=& 2\cos (k_x)-\cos \left(k_x/\phantom{k_{x}2}2+\sqrt{3}{k}_y/\phantom{\sqrt{3}{k}_{y}2}2\right)\\ & -\cos \left(k_x/\phantom{k_{x}2}2-\sqrt{3}{k}_y/\phantom{\sqrt{3}{k}_{y}2}2\right),\\ {\mathrm{\Delta }}_{E2\mathrm{N}\mathrm{N}}^{d_{xy}}(\mathit{k})=& -\cos (k_x)+2\cos \left(k_x/\phantom{k_{x}2}2+\sqrt{3}{k}_y/\phantom{\sqrt{3}{k}_{y}2}2\right)\\ & -\cos \left(k_x/\phantom{k_{x}2}2-\sqrt{3}{k}_y/\phantom{\sqrt{3}{k}_{y}2}2\right),\end{array}

(9)\begin{array}{rl}{\mathrm{\Delta }}_{E2\mathrm{N}\mathrm{N}\mathrm{N}}^{d_{x^2-y^2}}(\mathit{k})=& 2\cos (\sqrt{3}{k}_y)-\cos \left(3k_x/\phantom{3k_{x}2}2+\sqrt{3}{k}_y/\phantom{\sqrt{3}{k}_{y}2}2\right)\\ & -\cos \left(-3k_x/\phantom{-3k_{x}2}2+\sqrt{3}{k}_y/\phantom{\sqrt{3}{k}_{y}2}2\right),\\ {\mathrm{\Delta }}_{E2\mathrm{N}\mathrm{N}\mathrm{N}}^{d_{xy}}(\mathit{k})=& -\cos (\sqrt{3}{k}_y)+2\cos \left(3k_x/\phantom{3k_{x}2}2+\sqrt{3}{k}_y/\phantom{\sqrt{3}{k}_{y}2}2\right)\\ & -\cos \left(-3k_x/\phantom{-3k_{x}2}2+\sqrt{3}{k}_y/\phantom{\sqrt{3}{k}_{y}2}2\right),\end{array}

(10)\begin{array}{rl}{\mathrm{\Delta }}_{E2\mathrm{T}\mathrm{N}\mathrm{N}}^{d_{x^2-y^2}}(\mathit{k})=& 2\cos (2k_x)-\cos \left(k_x+\sqrt{3}{k}_y\right)\\ & -\cos \left(k_x-\sqrt{3}{k}_y\right),\\ {\mathrm{\Delta }}_{E2\mathrm{T}\mathrm{N}\mathrm{N}}^{d_{xy}}(\mathit{k})=& -\cos (2k_x)+2\cos \left(k_x+\sqrt{3}{k}_y\right)\\ & -\cos \left(k_x-\sqrt{3}{k}_y\right),\end{array}

(11)\begin{array}{rl}{E}_{\mathrm{S}\mathrm{C}}(\mathit{k})=& s_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}\sin \theta \cos \phi \left({\mathrm{\Delta }}_{E2\mathrm{N}\mathrm{N}}^{d_{x^2-y^2}}(\mathit{k})+\mathrm{i}{\mathrm{\Delta }}_{E2\mathrm{N}\mathrm{N}}^{d_{xy}}(\mathit{k})\right)\\ & +s_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}\sin \theta \sin \phi \left({\mathrm{\Delta }}_{E2\mathrm{N}\mathrm{N}\mathrm{N}}^{d_{x^2-y^2}}(\mathit{k})+\mathrm{i}{\mathrm{\Delta }}_{E2\mathrm{N}\mathrm{N}\mathrm{N}}^{d_{xy}}(\mathit{k})\right)\\ & +s_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}\cos \theta \left({\mathrm{\Delta }}_{E2\mathrm{T}\mathrm{N}\mathrm{N}}^{d_{x^2-y^2}}(\mathit{k})+\mathrm{i}{\mathrm{\Delta }}_{E2\mathrm{T}\mathrm{N}\mathrm{N}}^{d_{xy}}(\mathit{k})\right).\end{array}

Here, s_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=\sqrt{{(s_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}\sin \theta \cos \phi )}^2+{(s_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}\sin \theta \sin \phi )}^2+{(s_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}\cos \theta )}^2} denotes the total pairing strength. The polar angle θ ∈ [0, π/2] controls the ratio of TNN to in-plane pairing (NN and NNN), while the azimuthal angle φ ∈ [0, π/2] regulates the relative weight of NN and NNN pairings. The long-range pairing is thus parameterized by (s_total, θ, φ). Specific configurations include: pure NN pairing when θ = π/2 and φ = 0; pure NNN pairing when θ = π/2 and φ = π/2; and pure TNN pairing when θ = 0 (with φ arbitrary, though conventionally set to 0). And intermediate angles correspond to various mixed configurations. Within the Nambu basis \psi =(c_{k\uparrow },c_{k\downarrow },c_{-k\uparrow }^{†},c_{-k\downarrow }^{†}{)}^{\mathrm{T}}, the BdG Hamiltonian for the triangular lattice, incorporating hopping term E_t(k) and long-range pairing term E_sc(k), takes the following form:

(12)H(\mathit{k})=\left(\begin{array}{cccc}{E}_t(\mathit{k})-\mu & 0& 0& E_{\mathrm{S}\mathrm{C}}(\mathit{k})\\ 0& E_t(\mathit{k})-\mu & -E_{\mathrm{S}\mathrm{C}}(\mathit{k})& 0\\ 0& -E_{\mathrm{S}\mathrm{C}}^{†}(\mathit{k})& -E_t(-\mathit{k})+\mu & 0\\ E_{\mathrm{S}\mathrm{C}}^{†}(\mathit{k})& 0& 0& -E_t(-\mathit{k})+\mu \end{array}\right),

where the hopping term E_t includes the NN, NNN and TNN with parameters t₁, t₂ and t_3. In numeric, t = t₁ = 1.0 is set as unity and t₂ = 0.1t₁ and t₃ = 0.01t₁. E_SC(k) is the long-range pairing potential, including NN, NNN and TNN pairings.

To study topological properties of TSC phase, we initially employ efficient method [24] to compute CN as a function of chemical potential and long-range pairing strength. The CN can be given by

(13)CN={\displaystyle \frac{1}{2\pi \mathrm{i}}}{\displaystyle \sum _{k_{\ell }}\ln \left[U_1\left(k_{\ell }\right)U_2\left(k_{\ell }+\hat{1}\right)U_1{\left(k_{\ell }+\hat{2}\right)}^{-1}{U}_2{\left(k_{\ell }\right)}^{-1}\right],}

(14)U_{\nu }\left(k_{\ell }\right)={\displaystyle \frac{⟨n\left(k_{\ell }\right)|n\left(k_{\ell }+\hat{\nu }\right)⟩}{\left|⟨n\left(k_{\ell }\right)|n\left(k_{\ell }+\hat{\nu }\right)⟩\right|}},

where \hat{\nu } is a vector in the direction, k_{\ell } denotes lattice points on the discrete Brillouin zone. |n\left(k_{\ell }\right)⟩ is a wave function of the nth Bloch band.

Figures 1(c)−(e) show that as the pairing distance increases, the phase distribution of pairing function becomes more diverse and complex, and the positions of sudden phase jumps increase. In the absence of a long-range pairing potential, this studied model displays a trivial insulator as plotted in Figs. 1(f) and (g).

3 Results and discussion

The triangular lattice can host TSC states under long-range pairing even without spin–orbit coupling, as shown in Figs. 2(a)−(c). This offers a less constrained pathway to topological superconductivity compared to conventional requirements, which typically require the combined presence of superconductivity, strong spin–orbit coupling, and TRS breaking [9, 10, 27]. As the chemical potential μ varies, the band-gap closes and reopens, accompanied by changes in the CN, as illustrated in Figs. 2(a)−(c). This behavior aligns with the established understanding that topological phase transitions are driven by band gap closure and reopening [28]. Under the NN pairing, the studied system shows TSC state with a CN of –4 within the range μ ∈ [–6.6, 2.4]t. For NNN and TNN pairings [Figs. 2(b) and (c)], nonzero CNs of 4, –2, –8 and –4, 8 emerge with increasing μ, respectively. Notably, mixed pairings do not give rise to new CNs beyond those found in individual pairing channels. This suggests the absence of cooperative effects among NN, NNN, and TNN pairings — that is, no new CN phases emerge — and points instead to a competitive relationship among them. Among these, the TNN pairing dominates, as evidenced by its broad topological phase region in Figs. 2(d)−(f). This further confirms that long-range pairing can actively enhance topological superconductivity [18].

The Berry curvature, determined by the electronic band structure, can develop sharp peaks or valleys near momentum points where band inversion occurs [29]. In the presence of long-range pairing, TRS is broken, leading to distinct peaks or valleys in the Berry curvature distribution — a signature of nontrivial topology in the system [Figs. 3(e)–(j)]. When considering NN pairing, the band-gap along Γ–K is smaller than that along Γ–M, and band inversion tends to occur at the momentum path with a relatively small band-gap [Fig. 3(a)]. The Berry curvature displays peaks along the Γ–K high-symmetry line, indicating that band inversion occurs on this path [Fig. 3(e)]. For NNN and TNN pairings, both band structures and Berry curvature distributions indicate that band inversion takes place along the Γ–M and Γ–K lines, respectively [Figs. 3(b, c, f, g)]. Interestingly, as shown in Fig. 2, the CN does not vary with increasing long-range pairing strength, which differs from previous results [12]. This can be attributed to the fact that long-range pairing only alters the magnitude of band-gap and modifies slightly band shapes without inducing band-gap closure and reopening, thereby failing to drive a topological phase transition. Consistently, Figs. 3(h)–(j) show that the Berry curvature distribution evolves smoothly with increasing pairing strength, leaving the topological character of the bands unaffected. In mixed-pairing regimes such as (s_total, 20°, 10°) and (s_total, 60°, 10°), where NN and TNN components dominate, band inversion and Berry curvature features preferentially appear along the Γ–K path, where the band-gap remains relatively small [Figs. 3(d, k, l)]. This behavior aligns with the trends observed in pure NN and TNN pairings. It further corroborates that no new topological CNs emerge under mixed pairing, consistent with the results presented above in Figs. 2(d)–(f).

To further investigate the topological property of the TSC state, we construct a tight-binding model for an infinitely long strip of triangular lattice with finite width 600 along both zigzag and armchair edges [Fig. 1(a)]. The two energy bands near the E_F are doubly degenerate due to spin degeneracy as shown in Fig. 4 and Fig. 5. The morphology of topological edge states is strongly influenced by ribbon boundary symmetry, leading to distinct behaviors at different edges [30]. Based on the bulk analysis, band inversion occurs along the Γ–K and Γ–M high-symmetry paths (Fig. 3), which governs the location of topological edge states. Long-range pairing effectively introduces “long-range hopping” in real space, thereby modifying the edge band dispersion. The phase distribution of the NN pairing function in the first Brillouin zone is relatively smooth [Fig. 1(c)], whereas those of NNN and TNN pairings exhibit more abrupt variations [Figs. 1(d) and (e)]. Correspondingly, the Berry curvature distribution becomes more scattered [Figs. 3(e, f, h, i)], suggesting that topological edge states become relatively disordered and may interact with one another. As illustrated in Fig. 4, the zigzag-edge ribbon hosts topological edge states with clear particle−hole symmetry, and the number of edge state crossings agrees with the CN. In contrast, the armchair-edge ribbon exhibits different symmetry behavior: under NN pairing, particle-hole symmetry is preserved and the edge states align with the CN; however, under NNN and TNN pairings, particle-hole symmetry is broken, and the edge states no longer cross the E_F. Increasing the long-range pairing strength widens the edge band-gap without altering the topological character, which consistent with the bulk results in Fig. 2 and Fig. 3. For mixed pairings such as (s_total, 20°, 10°) and (s_total, 60°, 10°), the topological edge states resemble those of pure long-range pairings [Figs. 4(g) and (h)]. In zigzag ribbons, edge states remain robust and accurately reflect the CN. In armchair ribbons, broken particle–hole symmetry leads to asymmetric edge dispersions that avoid crossing the E_F [Figs. 5(g) and (h)]. No new types of edge crossings or degeneracies emerge beyond those found in individual pairing cases, reinforcing the competitive (rather than cooperative) interplay among pairing channels, as inferred from the bulk phase diagram in Fig. 2.

To comprehensive study of the topological edge state of the system, we calculate the |ψ(n)|² of zigzag [Figs. 6(a)–(f)] and armchair ribbons [Figs. 6(g)–(l)], which display topological superconductivity in real space near the E_F. Under varying pairing ranges and strengths, the real space |ψ(n)|² in zigzag and armchair ribbons exhibits pronounced edge localization, indicating that the Majorana zero modes are well-separated and confined to the boundaries — consistent with previous reports [17]. As the pairing function strength increases, the bulk superconducting gap of the system becomes larger, as shown in Figs. 3(a)–(c). A larger bulk band-gap can more effectively isolate the topologically protected edge states from the bulk states (see Figs. 4 and 5), helping to suppress the propagation of bulk states, making the topological edge states more localized, and thereby improving the purity and stability of the topological edge states. The enhancement of the localized characteristic of topological edge states also implies that the studied system exhibits stronger topological properties. This demonstrates that the effect of the pairing strength on both topological edge states and |ψ(n)|² is coherent and mutually consistent.

4 Conclusion

In conclusion, our study systematically investigates TSC states in a triangular lattice with long-range superconducting pairing, including NN, NNN, and TNN pairing. The spontaneous breaking of TRS under long-range pairing results in pronounced features in the Berry curvature and the emergence of non-zero CNs, confirming the coexistence of topological and superconducting properties. Variations in μ derive band-gap closures and re-openings accompanied by changes in the CN, consistent with bulk−boundary correspondence. Notably, the CN remains invariant with increasing long-range pairing strength, indicating that long-range pairing modifies the band-gap size without altering the topological nature of the bands, thereby confirming the robustness of long-range superconducting states. The study of topological edge state show that the NN pairing preserves the particle–hole symmetry of the topological edge state along the k_x and k_y directions, consistent with CN, whereas NNN and TNN pairings break this symmetry along the kₓ direction. This indicates that the boundary symmetry and the interaction range of long-range pairing functions play a decisive role in the distribution of topological electronic states. The analysis of the |ψ(n)|² near the E_F further indicates that as the pairing potential strength increases, the energy range of topological edge state become more wider, resulting in enhanced edge localization of the |ψ(n)|². Overall, our study proposes a TSC state realization mechanism that does not rely on spin–orbit coupling and magnetic field, providing a feasible strategy for the experimental design of topological superconductors, and reducing the strict requirements for material constraints.

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