Boundary obstructed topological superconductor in buckled honeycomb lattice under perpendicular electric field

TL;DR

This paper demonstrates that a buckled honeycomb lattice with f-wave spin-triplet pairing can host boundary-obstructed topological superconductivity, with stable corner modes under certain boundary conditions, expanding understanding of topological phases in such materials.

Contribution

It reveals that boundary-obstructed topological superconductivity can occur in buckled honeycomb lattices with fSTP, especially under nonzero sublattice potential, and links boundary modes to Kitaev chain physics.

Findings

Boundary-obstructed topological superconductor (BOTS) can exist in buckled honeycomb lattices.

Corner modes remain stable when the boundary is gapped.

Boundary modes in the normal state relate to Kitaev chain Hamiltonians.

Abstract

In this work, we show that a buckled honeycomb lattice can host a boundary-obstructed topological superconductor (BOTS) in the presence of f-wave spin-triplet pairing (fSTP). The underlying buckled structure allows for the manipulation of both chemical potential and sublattice potential using a double gate setup. Although a finite sublattice potential can stabilize the fSTP with a possible higher-order band topology, because it also breaks the relevant symmetry, the stability of the corner modes is not guaranteed. Here we show that the fSTP on the honeycomb lattice gives BOTS under nonzero sublattice potential, thus the corner modes can survive as long as the boundary is gapped. Also, by examining the large sublattice potential limit where the honeycomb lattice can be decomposed into two triangular lattices, we show that the boundary modes in the normal state are the quintessential ingredient leading to the BOTS. Thus the effective boundary Hamiltonian becomes nothing but the Hamiltonian for Kitaev chains, which eventually gives the corner modes of the BOTS.

Equations

$$H_{\text{BdG}}(\mathbf{k})=t_1{H}_1^{\text{++}}(\mathbf{k}){\sigma }_x{\tau }_z+t_1{H}_{1^{\prime }}^{\text{-+}}(\mathbf{k}){\sigma }_y{\tau }_z+t_2{H}_2^{\text{++}}(\mathbf{k}){\tau }_z$$

$$-\mu {\tau }_z+M{\sigma }_z{\tau }_z+{\Delta }_s{H}_3^{\text{+-}}(\mathbf{k})s_x{\tau }_y,$$

$$t_1{H}_1^{\text{++}}(\mathbf{k}){\sigma }_x+t_1{H}_{1^{\prime }}^{\text{-+}}(\mathbf{k}){\sigma }_y+H_3^{\text{+-}}(k){\Delta }_s{\tau }_z{\sigma }_z.$$

$$H_{\text{b}}(\theta ,k_{\parallel })=v_{\theta }{\tau }_z{s}_0{k}_{\parallel },$$

$$H_{\text{K}}(k)=H_{\text{N}}(k){\tau }_z+H_{\text{SO}}(k)s_z+H_{\text{R}}(k)s_x+H_{\Delta }(k)s_x{\tau }_y,$$

Full text

Introduction.— Although superconductivity has not yet been observed in pristine graphene, it has been experimentally realized in related families such as multi-layer [1, 2, 3], twisted moiré bi-layer [4], and alkali metal intercalated graphene [5, 6, 7, 8, 9]. Remarkably, underlying symmetries of the honeycomb lattice allow the emergence of various topologically nontrivial states, such as chiral p-wave and d-wave pairings [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Recently, it has been shown that the f-wave spin-triplet pairing (fSTP) [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38] can arise in the honeycomb lattice, deriving higher-order topological superconductivity (HOTS) [39, 40] that hosts zero-energy Majorana corner modes, protected by both bulk gap and certain spatial symmetries, such as inversion symmetry [41, 42, 43, 44, 45, 46, 47, 48, 45, 49, 50, 51, 52, 53]. Also, fSTP was shown to be enhanced when there is a large sublattice potential [54, 55]. For instance, an electric field perpendicular to the plane of a low-buckled honeycomb lattice such as silicene can induce a considerable sublattice potential and stabilize fSTP (see Fig. 1(b), and Fig. 1(c)) [56]. However, as the sublattice potential breaks the essential symmetries that protect HOTS simultaneously, the stability of the corner modes is not guaranteed unless a distinct mechanism for their intrinsic protection exists.

In this letter, we show that fSTP can host corner modes that remain intact even under considerable sublattice potential. Interestingly, the sublattice potential turns the HOTS into an extrinsic topological superconductor [41, 42, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69] or a boundary obstructed topological superconductor (BOTS) [70, 71, 72, 73, 74, 75, 76, 77], where corner modes are protected by the energy gap of the edge Hamiltonian, not the bulk Hamiltonian. We also propose a double gate setup that enables the realization of a tunable BOTS phase transition.

In the following, we first illustrate how unbroken symmetries can protect the BOTS phase, and then analyze the large sublattice potential limit of honeycomb lattices which effectively can be mapped into two triangular lattices. We demonstrate that the existence of boundary modes in the normal state (without superconductivity) is crucial for the BOTS (in the superconducting state). As fSTP acts effectively as a p-wave pairing for these boundary modes, the corner modes in the BOTS can be interpreted as the end modes of Kitaev chains [78] on the boundaries.

Model.— The Bogoliubov-de-Gennes (BdG) Hamiltonian for the honeycomb lattice with $p_{z}$ orbital and fSTP is given by (see Sec. S1 in the Supplemental Material (SM) for a derivation) 111 See Supplemental Material at **** for a derivation of the BdG Hamiltonian, a derivation of the edge Hamiltonian and symmetry representation, a derivation of the effective Hamiltonian in the large sublattice limit, a discussion on the spin-polarized case, a discussion on BOTS in large spin-orbit coupling, a discussion on the emergence of extrinsic HOTS by employing the valley hall effect, numerical calculation of topological invariant for the slab of the honeycomb lattice, and study of the BOTS with different edge termination.

[TABLE]

where $t_{1}$, $t_{2}$, $\mu$, and $M$ are nearest-neighbor hopping, next-nearest-neighbor hopping, chemical potential, and sublattice potential, respectively  [80, 81]. We note that $\mu$ and $M$ can be changed by controlling the gate voltage (see Fig. 1(b)). In Eq. (Boundary obstructed topological superconductor in buckled honeycomb lattice under perpendicular electric field), $\sigma_{x,y,z}$, $s_{x,y,z}$, $\tau_{x,y,z}$ are the Pauli matrices for sublattice, spin, and electron-hole degrees of freedom. The $H^{\eta_{1},\eta_{2}}_{n}(k_{x},k_{y})$ are momentum dependent functions, in which $H^{\eta_{1},\eta_{2}}_{n}(k_{x},k_{y}){=}\eta_{1}H^{\eta_{1},\eta_{2}}_{n}(-k_{x},k_{y})$, $H^{\eta_{1},\eta_{2}}_{n}(k_{x},k_{y}){=}\eta_{2}H^{\eta_{1},\eta_{2}}_{n}(k_{x},-k_{y})$ and their explicit forms are provided in the SM [79]. In Eq. (Boundary obstructed topological superconductor in buckled honeycomb lattice under perpendicular electric field), $\Delta_{s}{=}(\Delta_{A}{+}\Delta_{B})/2$ is the sublattice symmetric amplitude of fSTP, where $\Delta_{A}$ and $\Delta_{B}$ are the amplitude of fSTP on A and B sublattices, respectively. The $H^{\text{+-}}_{3}(\mathbf{k})$ in Eq. (Boundary obstructed topological superconductor in buckled honeycomb lattice under perpendicular electric field) is the Fourier transform of fSTP in the real space (see arrows inside Fig. 1(a)), in which its zeros (pairing nodes) are located along $\Gamma-M_{1,2,3}$ (see Fig. 1(d)). In this letter, we are interested in a low doping limit in which the fSTP always gives a fully gapped superconductor to distinguish the in-gap corner modes. In the following, to investigate the topological state of Eq. (Boundary obstructed topological superconductor in buckled honeycomb lattice under perpendicular electric field), we suppose a nonzero fSTP regardless of $\mu$ and $M$.

Results.— We summarize the topological characteristic of Eq. (Boundary obstructed topological superconductor in buckled honeycomb lattice under perpendicular electric field) in the last row of Fig. 2. In Fig. 2(a), we show that when $\mu{=}M{=}0$ (we set $t_{2}{=}0$ for simplicity), the system hosts gapless states for all boundaries. In Fig. 2(b), we turn on $\mu{\neq}0$ and put $M{=}0$. In this case, the gapless edge modes still exist along the armchair edges (Fig. 2(b4) and top and bottom edges of Fig. 2(b2)). Here, the system is a topological crystalline superconductor, where $\mathcal{C}_{2y}{:}(x,z){\rightarrow}{-}(x,z)$ symmetric edges (armchair edges) remain gapless. In the absence of these symmetric edges, the system is HOTS (see Fig. 3(f)), which has been already discussed in Ref. 40. The HOTS arises from the fact that certain symmetry operators (e.g. inversion) flip the gap sign on the edge Hamiltonian, which gives in-gap states, called corner modes. In Fig. 2(c) we put $0{<}|M|{<}|\mu|$. In this case, although no gapless one-dimensional edge mode remains for the system, zero-energy corner modes still appear. Note that depending on the signs of $M$ and $\mu$, the corner modes appear at different edges. We will show that this phase is a BOTS. The BOTS is protected by mirror $\mathcal{M}_{y}:y\rightarrow-y$ symmetry and corner modes survive as long as $\mathcal{M}_{y}$ symmetric edges (zigzag edges) remain gapped. In Figs. 2(d)-(e), we show that corner modes can be eliminated via a gap closing and reopening at one of the zigzag edges (right edge of Fig. 2(d2) and edge-2 in Fig. 2(d3)), which we call BOTS phase transition. Note that the bulk remains gapped during this transition (black part of the energy spectrum in the third and fourth rows of Fig. 2).

Symmetries Equation (Boundary obstructed topological superconductor in buckled honeycomb lattice under perpendicular electric field) has particle-hole symmetry $\Xi{=}\tau_{x}\mathcal{K}{}_{[\mathbf{k}{\rightarrow}{-}\mathbf{k}]}$, and time-reversal symmetry $\Theta{=}s_{y}\tau_{z}\mathcal{K}{}_{[\mathbf{k}{\rightarrow}{-}\mathbf{k}]}$, where $\mathcal{K}$ is the complex conjugate operator. The system is also invariant under a $U(1)$ spin rotation, whose generator is given by $\mathcal{S}{=}s_{z}\tau_{z}$. Moreover, Eq. (Boundary obstructed topological superconductor in buckled honeycomb lattice under perpendicular electric field) is invariant under a three-fold rotational symmetry $\mathcal{C}_{3z}{=}e^{i\tfrac{\pi}{3}s_{z}\tau_{z}}{}_{[\mathbf{k}{\rightarrow}\mathcal{R}_{2\pi/3}\mathbf{k}]}$, and mirror symmetry $\mathcal{M}_{y}{=}s_{y}{}_{[{k_{y}\rightarrow}{-}k_{y}]}$, where $\mathcal{R}_{\theta}$ rotates momentum vector $\mathbf{k}$ by $\theta$ around the out-of-plane direction. Furthermore, when $M{=}0$, Eq. (Boundary obstructed topological superconductor in buckled honeycomb lattice under perpendicular electric field) has inversion symmetry $\mathcal{P}{=}\tau_{z}\sigma_{x}{}_{[\mathbf{k}{\rightarrow}{-}\mathbf{k}]}$, a six-fold rotational symmetry $\mathcal{C}_{6z}{=}e^{i\tfrac{\pi}{6}s_{z}\tau_{z}}\sigma_{x}\tau_{z}{}_{[\mathbf{k}{\rightarrow}\mathcal{R}_{2\pi/6}\mathbf{k}]}$, and a two-fold rotational symmetries $\mathcal{C}_{2y}{=}s_{y}\sigma_{x}\tau_{z}{}_{[k_{x}{\rightarrow}{-}k_{x}]}$. When $\mu{=}M{=}t_{2}{=}0$, Eq. (Boundary obstructed topological superconductor in buckled honeycomb lattice under perpendicular electric field) additionally has a local symmetry $\mathcal{L}{=}s_{x}\sigma_{z}\tau_{y}$ that commutes with $\mathcal{S}$.

First order topological phase As we have shown in Fig. 2(a), when $\mu{=}M{=}t_{2}{=}0$, the system supports gapless edge modes at all edges, implying the existence of a first-order topological phase. To see this, it is enough to rewrite $H_{\text{BdG}}(\mathbf{k})$ using a unitary matrix $U$ (given in the SM [79]) in the basis where both $U^{\dagger}\mathcal{S}U{=}s_{z}$ and $U^{\dagger}\mathcal{L}U{=}\tau_{z}$ (and $U^{\dagger}\sigma_{z}U{=}\sigma_{z}$) are diagonal and $U^{\dagger}H_{\text{BdG}}(\mathbf{k})U{=}$

[TABLE]

Therefore, in each sector of $s_{z}{=}\pm 1$, $H_{\text{BdG}}$ decomposes to two copies of the celebrated Haldane model [82] with opposite Chern numbers; similar to the two-dimensional quantum spin Hall insulator model proposed in graphene [83, 84].

Edge theory Here, we derive an edge Hamiltonian at the boundary of a disk geometry to study the effect of nonzero $\mu$, $M$, and $t_{2}$ on the gapless boundary modes. The edge Hamiltonian of Eq. (2) composed of two counterpropagating chiral edge states, can be written as

[TABLE]

where $v_{\theta}$ is the Fermi velocity of the gapless edge modes that depends on the polar angle $\theta$, and $k_{\parallel}$ is the locally defined momentum tangent to the disk boundary (see Fig. 3(a)). Note that the edge Hamiltonian has to satisfy all symmetries of Eq. (2), whose representations can be obtained by considering their definitions regarding the edge Hamiltonian and commutation or anti-commutation relations between them (see Sec. S2 of SM for a derivation [79]). The representations of particle-hole, time-reversal, $\mathcal{L}$, and $\mathcal{S}$ symmetries are given by $\Xi{=}s_{x}\tau_{z}\mathcal{K}{}_{[k_{\parallel}{\rightarrow}{-}k_{\parallel}]}$, $\Theta{=}is_{y}\tau_{x}\mathcal{K}{}_{[k_{\parallel}{\rightarrow}{-}k_{\parallel}]}$, $\mathcal{L}{=}\tau_{z}$, and $\mathcal{S}{=}s_{z}$, respectively. For spatial symmetries, we find that there are two types of symmetry operations: type-1 symmetries (proportional to $\tau_{y}$ or $\tau_{z}$, such as $\mathcal{C}_{6z}$, $\mathcal{C}_{2y}$, and $\mathcal{P}$), and type-2 symmetries (proportional to $\tau_{x}$ or $\tau_{0}$, such as $\mathcal{C}_{3z}$, and $\mathcal{M}_{y}$), where type-1 symmetries exchange the sublattice index, whereas type-2 symmetries preserve it.

Possible edge mass We can study the topological feature of this system by understanding how symmetry operations keep edge Hamiltonian gapless [85]. The only edge mass that anticommutes with Eq. (3), and respects both $\Xi$ and $\Theta$ is $H_{\text{b.m}}(\theta)=m(\theta)\tau_{x}$. However, since $\tau_{x}$ is odd under $\mathcal{L}$, edge Hamiltonian remains gapless ($m(\theta){=}0$) if $\mathcal{L}$ is present (i.e. $\mu{=}M{=}t_{2}{=}0$). On the other hand, spatial symmetries relate edge masses at different $\theta$. If we add a mass term that respects certain symmetries to the bulk BdG Hamiltonian, we expect it gives an edge mass that also respects the same symmetries. The mass terms that are invariant under type-1 symmetries give $m(\theta_{\text{p}}){=}{-}m(\theta)$, where $\theta_{\text{p}}$ means the symmetric partner of $\theta$ related by the given symmetry. Meanwhile, the mass terms that are invariant under type-2 symmetries give $m(\theta_{\text{p}}){=}{+}m(\theta)$.

HOTS Let us first assume that the system is invariant under type-1 symmetries, which force mass sign changing for symmetry related edges, giving zero energy corner modes [86] (see Fig. 3(b)). Then, the system belongs to HOTS [85, 87]. We note that for the symmetric edges where $\theta_{\text{p}}{=}\theta$, such as $\theta{=}\tfrac{\pi}{2}$ (armchair edges) under $\mathcal{C}_{2y}$, the symmetry operation forces $m(\theta_{\text{p}}{\equiv}\theta){=}{-}m(\theta)=0$. Hence, the gapless boundary modes of the first-order topological phase are protected by symmetry operators on the symmetric edges. Accordingly, we can call this phase a $\mathcal{C}_{2y}$ protected topological crystalline superconductor [88]. Both $\mu$ and $t_{2}$ are invariant under type-1 symmetries and keep the gapless states of the armchair edges (Fig. 2(b2) and Fig. 2(b4)), while gap out gapless states of zigzag edges (Fig. 2(b2) and Fig. 2(b3)). Thus, the hexagonal geometry with only zigzag edges hosts only corner modes (see Fig. 3(f)).

BOTS Now let’s assume that the system is only invariant under type-2 symmetries, which do not force any topological gapless states, as symmetry-related edges obtain the same mass sign. Despite this fact, the system can still have two different phases that cannot be smoothly connected without a gap closing at a symmetric edge (see Fig. 3(c)-(e)) [70, 71]. For instance, $\mu$ and $M$ (and $t_{2}$) are invariant under type-2 symmetries. Therefore, the HOTS (when $\mu{\neq}0$, $M{=}0$) can be smoothly transformed to BOTS by adding tiny $|M|$. Accordingly, although small $M$ trivialize HOTS, still in-gap corner modes are protected because of the underlying BOTS phase (Fig. 2(c2) or Fig. 3(f)). These corner modes can be eliminated only by a BOTS phase transition through a gap closing at a symmetric edge (the zigzag edge located at the right side of Fig. 2(d2) when $|M|{\approx}|\mu|$).

BOTS Phase transition We can obtain a rough estimation of the BOTS transition for geometries with only zigzag edges. For instance, consider the hexagonal geometry given in Fig. 3(f), where it has six edges that are labeled by ei ({i=1,…,6}) and their corresponding edge-masses are denoted by $m_{ei}$. We first obtain the edge mass at one of the edges (for instance e4) and obtain others by symmetry considerations. To obtain $m_{e4}$, we need to find zero energy modes at that edge (when $\mu{=}t_{2}{=}M{=}0$) and then project $\mu$, $t_{2}$, and $M$ onto these zero energy modes [89], which leads to $m_{e4}{=}{-}\mu{-}2t_{2}{+}M$ (see Sec. S3 in the SM [79]). The anti-commutation of a mass term (i.e. $M$) with type-1 symmetries forces $m(\theta_{\text{p}}){=}m(\theta)$. Thus, we obtain $m_{e3}{=}\mu{+}2t_{2}{+}M$, where signs of $\mu$ and $t_{2}$ are changed but $M$ keeps its sign. Using type-2 symmetries, the e4, e1, and e5 (e2, e3, and e6) edges obtain the same edge mass. As a result, nearby edges of Fig. 3(f) obtain masses with opposite signs when $|M|<|\mu+2t_{2}|$ and give BOTS transition at $|M|=|\mu+2t_{2}|$, which leads to a trivial phase by further increasing $|M|$.

Large sublattice potential limit In the large $M$ limit, wave functions corresponding to the positive and negative energy bands of the honeycomb lattice are polarized on one sublattice (see Fig. 4(b)). Therefore in this regime, the honeycomb lattice can be effectively decomposed into two triangular lattices, which we denote by $\kappa{=}{\pm}1$ (see Fig. 4(a)). For instance, if we first take into account only $t_{1}$ and $M$, we can write an effective Hamiltonian on the two triangular lattices with nearest-neighbor hopping $t^{\prime}{=}\tfrac{t_{1}^{2}}{2\kappa M}$ and onsite potential $V^{\prime}{=}\kappa M{+}\tfrac{zt_{1}^{2}}{2\kappa M}$ (see Sec. S4 in SM [79]). Here $z$ is the coordination number of a site that counts the number of the honeycomb links connected to it. When $|M{|\gg}|t_{1}|$, we confirm that this effective Hamiltonian (by choosing $z$=3 for all sites) gives the identical bulk energy spectrum to the honeycomb lattice. However, the zigzag boundary modes [90, 91] in the normal state of the honeycomb lattice are absent in the effective triangular lattices (compare Fig. 4(b), and (c)). As we have shown in Fig. 4(d), we can restore the normal state boundary modes in the triangular lattices by modifying the effective onsite potential considering the fact that at the zigzag boundary vertices $z$=2 (see Fig. 4(e)). Interestingly, turning on fSTP in the triangular lattice does not lead to any corner modes unless we modify the onsite potential at boundary vertices. Therefore, we can conclude that BOTS in the honeycomb lattice or modified triangular lattice are related to the presence of these normal state boundary modes.

Effective Kitaev-model and topological invariant Because normal state boundary modes are effectively localized at the boundary vertices [92], we can write an effective 1D Hamiltonian [93, 94], in which the fSTP acts as a p-wave pairing for these vertices (see Fig. 4(f)). Interestingly, this model can be considered as a generalization of the celebrated Kitaev model, hosting Majorana end modes [78, 95, 96, 97]. We can write a general effective Kitaev model including hopping, spin-orbit coupling (SOC), $M$, and pairing at the zigzag edges as

[TABLE]

where we apply Fourier transformation using $k$, the momentum parallel to the chain. In Eq. (4) $H_{\text{N}}(k)=-\mu+\kappa M+2t_{2}\cos(k)+F^{\kappa}(k)$, $H_{\text{SO}}(k)=2\kappa\lambda_{\text{SO}}\sin(k)$, $H_{\text{R}}(k)=2\lambda_{\text{R}}\sin(k)$, $H_{\Delta}(k)=2\Delta_{\kappa}\sin(k)$ and $F^{\kappa}(k)=\sum_{n=0}^{n_{c}}t_{n}^{\prime\kappa}\cos(nk)$, where $\lambda_{\text{SO}}$, $\lambda_{\text{R}}$, and $t_{n}^{\prime\kappa}$ are intrinsic SOC, Rashba SOC, and the effective hopping terms mediated by other (e.g. bulk) vertices, respectively. For instance, it is possible to write $F^{\kappa}(k){\approx}\tfrac{(1{+}\cos(k))t_{1}^{2}}{\kappa M}$ by employing the effective Hamiltonian of the triangular lattice derived in the previous section as the simplest approximation. The topological state of Eq. (4), is protected by both particle-hole $\Xi{=}\tau_{x}\mathcal{K}{}_{[k{\rightarrow}{-}k]}$, and time-reversal $\Theta{=}{s}_{y}\tau_{z}\mathcal{K}{}_{[k{\rightarrow}{-}k]}$ symmetries (see also Sec. S5 and Sec. S9 in the SM [79]), where the system can have a nontrivial $Z_{2}$ topological phase and the Majorana end modes come in pairs [96]. Note that when $\lambda_{R}{=}\lambda_{\text{SO}}{=}0$, Eq. (4) is invariant under $\tilde{\mathcal{M}}_{y}=\tau_{z}{}_{[k{\rightarrow}{-}k]}$, which acts effectively as an inversion symmetry for the Kitaev chain. Therefore, we can use the parity information $\xi(k)$ at $k{=}0,\pi$ to obtain $Z_{2}=\prod_{k=0,\pi}^{\prime}\xi(k)$ invariant, where ′ means the multiplication is done for only one state of the Kramer’s pair [96]. Because at $k=0,\pi$, Eq. (4) reduces to $H_{N}(k)\tau_{z}$, we only need to check the sign changing between $H_{N}(0)$ and $H_{N}(\pi)$. Therefore, we obtain $Z_{2}{=}\kappa\text{sgn}({-}\mu{+}\kappa M{-}2t_{2})$, and the topological transition occurs at $|M|{=}|\mu{+}2t_{2}|$, consistent with the edge state analysis. In the large $M$, normal state of two Kitaev chains $\kappa=\pm 1$ are energetically separated and accordingly for the geometry given in Fig. 4(f), only one of $\kappa=\pm 1$ can be topological at most. Hence, we can interpret Fig. 2(c2) containing a Kitaev chain with a topologically nontrivial phase on the right zigzag edge. The introduction of SOC does not break the nontrivial $Z_{2}$ topology of the effective Kitaev chain, as it preserves time-reversal symmetry. However, the presence of SOC can alter the equation defining the phase boundary, deviating slightly from $|M|=|\mu+2t_{2}|$, due to the emergence of other effective terms (see Sec. S7 in the SM [79]).

Note that in Eq. (4), considering sublattice dependent pairings $\Delta_{\kappa=\pm}{=}\Delta_{A,B}$, does not affect the topological properties of the effective Kitaev chains. Therefore, the presence of nonzero asymmetric fSTP $\Delta_{a}=\tfrac{\Delta_{A}{-}\Delta_{B}}{2}$ in the honeycomb lattice, resulting from an imbalance between the local density of states in the A and B sublattices under a nonzero sublattice potential, cannot alter the phase transition. Furthermore, the dependence of the BOTS on the boundary modes in the normal states allows controlling the Majorana fermions using boundary engineering (see Sec. S10 in the SM [79]). Additionally, a nontrivial topological state in the normal state always guarantees the existence of boundary modes and can lead to Majorana corner modes in the presence of fSTP (see Sec. S7 and Sec. S8 of the SM [79] for quantum spin hall effect and quantum valley hall effect [98], respectively).

Conclusion.— In this letter, we have studied the topological features of the f-wave superconductor in buckled honeycomb structures with sublattice potential. We first uncovered that despite breaking several symmetries by the sublattice potential, the remaining symmetries still protect BOTS with corner modes, and these corner modes disappear after a BOTS phase transition, mediated by gap closing on the symmetric edges. Furthermore, we uncover that generating boundary modes in the normal state (using boundary engineering or manipulating gate voltages) is an important step to realizing BOTS in the presence of fSTP. Our study can be extended to many recently discovered superconductors [1, 2, 3, 4] in which superconductivity may obtain an fSTP symmetry.