First- and higher-order topological superconductors on a square lattice

Abstract

Topological systems with hybrid topology offer unique opportunities for exploring multiplexing topological phenomena and valuable applications. However, building a hybrid topological superconductor and achieving a controllable topological phase transition between different orders of topological superconductors remain a challenge. We propose a solution to unify both first- and higher-order topological superconductors on a square lattice, incorporating the Rashba spin−orbit coupling, Zeeman field and s-wave superconducting pairing. By utilizing one-dimensional normal edge states, we construct a boundary-obstructed topological superconductor associated with closing the boundary energy gap. This leads to the emergence of Majorana corner modes, whose topological properties are characterized by the Berry phase. By tuning the amplitude of different intracell hoppings, we can control the localization of Majorana corner modes. We also generalize the Majorana polarization as a topological invariant to verify the existence of Majorana corner modes. Remarkably, the obtained phase diagram is well consistent with that described by the boundary energy gap, a quantized Berry phase and Chern number. With the further increase of Zeeman field, we observe a transition from a second- to first-order topological superconducting phase by closing the bulk gap. Its topology is protected by the bulk states and characterized by nonzero Chern number. Additionally, no Majorana corner modes are present and the topological boundary states are determined by nonzero Chern number in the region where both the quantized Berry phase and Chern number are nonzero. Furthermore, we achieve the hinge Majorana zero modes in a three-dimensional structure by stacking the two-dimensional square lattices. Our work unveils the physical mechanism to get a topological superconductor with different orders, and opens an avenue to characterize and detect different order topological superconductors on two-dimensional lattices.

Graphical abstract

Keywords

topological superconductors / hybrid topology / topological phase transition / Majorana corner modes / square lattice

1 Introduction

Topological superconductors (TSCs) hosting Majorana zero modes (MZMs) are of high interest due to their exceptional properties [1–9] and potential for applications in quantum information technologies [10–13]. To date, topological superconductivity has been studied in intrinsic unconventional superconductors [14–20] and various heterostructures where superconductivity is induced by the proximity effect [21–25]. With the advent of higher-order topological states, which generalize the standard bulk-boundary correspondence, various new platforms are proposed to realize MZMs. [26–40]. Generally, an nth-order topological phases in d spatial dimensions harbors gapless states on a (d- n) dimensional boundary, whereas first-order (conventional) TSCs correspond to the case with n=1. In this endeavor, MZMs are supposed to be localized at the corners of a two-dimensional (2D) second-order TSC [28–33, 40, 41].

Recently, a topological system with different topological classes have attracted increasing attention due to multiplexing topological phenomena and enhanced information processing. For example, a multi-gap topological photonic system with different topological classes has been realized, where the interactions between the topological edge states of distinct nature can enable nonlinear switch of photonic flows in the topological edge channels [42]. In a 2D nonsymmorphic metacrystal, Zhang et al. [43] realized the one-dimensional (1D) interface states and 0-dimensional (0D) corner states in two bulk band gaps, corresponding to the dipole and quadrupole topologies, respectively. Yang et al. [44] have also reported that an acoustic topological insulator with coexisting of the first-order and second-order topologies offers 1D helical edge states and 0D corner states, which could be used as bifunctional devices for sensing and waveguiding. In a three-dimensional (3D) insulator phase, Kooi et al. [45] proposed that the 2D gapless surface states and 1D hinge states can coexist in a bulk gap by breaking the translational symmetry in a weak topological insulator. Interestingly, the higher-order topological Weyl semimetals can also be proposed to host the Weyl points, Fermi arcs, and the 1D hinge states [46].

Additionally, research has been conducted on topological superconductors with different topological classes, which is not only of fundamental interest but also important for harnessing the advantages of both Majoarana corner and edge states when designing bifunctional devices [47–52]. In a 2D p-wave superconductor hosting a first-order topological state and existing in either a chiral or helical phase, the introduction of a d-wave superconducting pairing can induce zero-energy Majorana corner modes (MCMs) or partially gapped out edge states by tuning the parent p-wave phase [47]. Marsiglio et al. [48] unify both first- and higher-order topological superconductors in a extended Hubbard model with both Rashba and Dresselhaus spin−orbit coupling (SOC). In kagome quantum spin-3/2 liquids with tunable nearest-neighbor couplings, coexisting topologies exhibit, and the Majorana edge states can be tuned to localize towards one of the corners of a finite lattice [49]. Wu et al. [50] have also proposed a Floquet engineering scheme to generate 2D hybrid-order topological superconductors with both corner states and gapless chiral edge states. Besides the above proposals, it is important to explore alternative schemes for constructing a TSC with both first- and higher-order topology only using s-wave pairing, as s-wave superconductivity is more common and easier to achieve experimentally.

Generally, there are two types of phase transitions to higher-order topological superconductors (HOTSCs) that have been achieved. The type-I phase transition to HOTSCs involves the closing of the bulk energy gap. The topology of these superconductors is determined by topological invariants defined on the bulk bands [28–32, 38, 40, 48, 53]. The type-II phase transition to HOTSCs or to “boundary-obstructed topological superconductors” involves the closing of the energy gap at the boundaries [33, 41, 53, 54]. These boundary signatures are not related to bulk properties, and they are characterized by topological invariants defined on boundary bands [55]. We term the latter case a boundary-obstructed TSC. Because the dimension of the boundary-obstructed TSC is lower than the bulk by one, the whole system is a second-order TSC [13].

In this work, we propose a model for designing first and second-order TSCs on a 2D square lattice by controlling the topology of the edge and bulk states. The model can be realized in a simple 2D square lattice with Rashba SOC, Zeeman field and s-wave pairing. In the 2D lattice, each unit cell contains four sites (a,b,c,d), and a flux of \pi piercing each unit cell results in the appearance of 1D normal edge states. By making use of these 1D edge states, we construct a boundary-obstructed TSC where the bulk states are trivial. Notably, the MCMs localize at the intersection of 1D nontrivial/trivial edges, whose topological properties are characterized by the Z_2 topological invariant known as the Berry phase. By tuning the topology of different 1D edge states, we can control the localization of MCMs. To verify the existence of MCMs, we also generalize the Majorana polarization (MP) as a topological invariant. The derived phase diagram is well consistent with that obtained by the boundary energy gap, Berry phase and Chern number. As we increase the Zeeman field further, we observe a transition from a second- to first-order TSCs by closing the bulk gap. The topology of this transition is protected by the bulk states and is characterized by nonzero Chern number. Therefore, we unify both first- and higher-order TSCs on a 2D lattice by manipulating the topology of edge and bulk states, respectively. Moreover, our scheme is based on s-wave superconductivity, which is more common and easier to achieve experimentally than unconventional pairings. The tunability between MCMs and chiral Majorana edge states could promote the design of bifunctional devices and enhance information processing.

The paper is organized as follows. In Section 2, the TSC model Hamiltonian is introduced. Section 3 contains the results and discussions. Firstly, we clarify the physical mechanism behind achieving both first- and higher-order TSCs on a 2D lattice. We use the concepts of Berry phase and nonzero Chern number to characterize TSCs of different orders, and establish the bulk−boundary correspondence. Secondly, we generalize the concept of MP as a topological invariant in 2D structure, which will be helpful in studying MCMs. Thirdly, we present a topological phase diagram as a function of chemical potential and Zeeman field, demonstrating the robustness of MCMs against disorders within a wide range of disorder strengths. Lastly, we achieve hinge MZMs in a 3D structure by stacking our 2D model. In Section 4, we provide concluding remarks on our results.

2 Model

We start with a Rashba SOC coupled 2D square lattice model for the TSC hosting chiral Majorana edge modes from Ref. [56]. The Hamiltonian reads H_{2D} =H_0+H_{\mu } +H_{SO}+H_Z+H_{\mathrm{\Delta }}:

(1)\begin{array}{rl}& H_0=t\sum _{i\sigma }\sum _{\hat{m}=\hat{x},\hat{y}}(f_{i+\hat{m}\sigma }^{†}{f}_{i\sigma }+f_{i-\hat{m}\sigma }^{†}{f}_{i\sigma }),\\ & H_{\mu }=\mu \sum _{i\sigma }{f}_{i\sigma }^{†}{f}_{i\sigma },\\ & H_{SO}=-\lambda \sum _i[(f_{i-\hat{x}\downarrow }^{†}{f}_{i\uparrow }-f_{i+\hat{x}\downarrow }^{†}{f}_{i\uparrow })\\ & +i(f_{i-\hat{y}\downarrow }^{†}{f}_{i\uparrow }-f_{i+\hat{y}\downarrow }^{†}{f}_{i\uparrow })+H.c.],\\ & H_Z=h\sum _i(f_{i\uparrow }^{†}{f}_{i\uparrow }-f_{i\downarrow }^{†}{f}_{i\downarrow }),\\ & H_{\mathrm{\Delta }}=\mathrm{\Delta }\sum _i(f_{i\uparrow }^{†}{f}_{i\downarrow }^{†})+H.c.,\end{array}

where f_{i\sigma }^{†}(f_{i\sigma }) denotes an electron creation (annihilation) operator with spin (\sigma = \uparrow ,\downarrow) at site i=(i_x, i_y), and \hat{x}(\hat{y}) is a unit vector along the x(y) axis. \mu, h and \lambda denote the chemical potential, the Zeeman field along the \hat{z}-direction and Rashba SOC, respectively. \mathrm{\Delta } is the s-wave superconducting pairing amplitude.

To obtain MCMs and sufficient degrees of freedom to manipulate the topological behaviors, we restructure the square lattice as depicted in Fig.1(a), consisting of four sites (a,b,c ,d) in each supercell. Introducing asymmetric intracell hopping is a commonly used methods to achieve a second-order topological insulator [57, 58]. The intercell hopping amplitude, denoted as t^{\mathrm{\prime }}, and the four intracell hopping amplitudes, controlled by {\delta }_{1,2,3,4}, allow us to manipulate the topological behaviors. The negative signs of -t-{\delta }_4 and -t^{\mathrm{\prime }} between sites b and c are a gauge choice for a \varphi = \pi flux through each supercell [59]. Consequently, the H_0 in Eq. (1) is rewritten as

(2)\begin{array}{rl}{H}_0=& \sum _{i\sigma }[(t+{\delta }_1)a_{i\sigma }^{†}{c}_{i\sigma }+(t+{\delta }_3)a_{i\sigma }^{†}{d}_{i\sigma }+(t+{\delta }_2)b_{i\sigma }^{†}{d}_{i\sigma }\\ & +(-t-{\delta }_4)b_{i\sigma }^{†}{c}_{i\sigma }+t^{\mathrm{\prime }}{c}_{i\sigma }^{†}{a}_{i+\hat{x}\sigma }+t^{\mathrm{\prime }}{b}_{i\sigma }^{†}{d}_{i+\hat{x}\sigma }\\ & +t^{\mathrm{\prime }}{d}_{i\sigma }^{†}{a}_{i+\hat{y}\sigma }-t^{\mathrm{\prime }}{b}_{i\sigma }^{†}{c}_{i+\hat{y}\sigma }+H.c.],\end{array}

where i denotes the supercell site. The other terms in Eq. (1) remain unchanged. When {\delta }_{1,2,3,4}=0 and the flux \varphi =0, Eq. (2) reduces to Eq. (1). To distinguish different edges, we label the four edges of the square lattice as I, II, III and IV [Fig.1(a)]. In our modified 2D lattice structure, a flux \varphi =\pi induces the emergence of 1D normal edge state that separates from the bulk states. As shown in Fig.1(b), we present the band structure with open boundary conditions (OBCs) along the \hat{x} direction and periodic boundary conditions (PBCs) along the \hat{y} direction. Given {\delta }_3=0.3 and {\delta }_{1,2,4}= 0, the 1D gapped normal edge states, indicated by the red bands, appear solely along edge-IV, while the other three edges lack such states. By manipulating {\delta }_{1,2,3,4}, we can similarly obtain 1D normal edge states along the other edges. Through fine-tuning of the topology of the edge and bulk states, we can achieve a boundary-obstructed TSC with MCMs and a first-order TSC with chiral Majorana edge modes, respectively.

3 Results and discussion

3.1 The phase transition between second- and first-order TSCs

We first discuss how to achieve a boundary-obstructed TSC. By adjusting the parameters {\delta }_{1,2,3,4}, we can get a 1D gapped normal edge state along the corresponding edge. Without loss of generality, let’s assume {\delta }_3=0.3 and {\delta }_{1,2,4}= 0. In this case, a 1D gapped normal edge state emerges along the edge-IV, while the other three edge states merge into the bulk bands [Fig.1(b)]. Although the band topology of the bulk is trivial, we can construct a boundary-obstructed TSC based on the 1D gapped normal edge states. The procedure follows the widely known mechanism in superconducting nanowire systems [60, 61]: The Rashba SOC lifts spin degeneracy by displacing the parabolic bands horizontally in opposite directions. Then the Zeeman field lifts the remaining spin degeneracy at k_y=0 by breaking time reversal symmetry, opening up a gap of value 2h. When the chemical potential \mu lies in the Zeeman gap, the topological superconducting phase emerges along the edge-IV by introducing s-wave pairing. On the other hand, the remaining three edges exhibit trivial behavior.

In Fig.2, we present the existence of a boundary-obstructed TSC accompanying with a phase transition between the second- and first-order TSC. We fix the chemical potential at \mu =1 and illustrate the energy gap as a function of the Zeeman field h under PBCs in the \hat{x} and \hat{y} directions, respectively. As h increases, the band gap \mathrm{\Delta } E_x along the \hat{x} direction decreases and eventually closes at k_x= \pi after surpassing the critical value h_{c2} =1.73 [Fig.2(a)]. This critical value h_{c2} signifies the closing of the bulk gap. For h> h_{c2}, \mathrm{\Delta } E_x remains at zero continuously. This phenomenon can be explained by the emergence of a gapless chiral edge mode with linear dispersion, which is localized on the x edge and appears within the bulk energy gap. As a result, the system undergoes a first-order topological superconducting phase transition. Simultaneously, the band gap \mathrm{\Delta }{E}_y along the \hat{y} direction experiences a series of closing, reopening, and closing once more [Fig.2(b)]. This indicates the presence of two distinct types of topological phase transitions. The system successively transitions from a trivial phase () to a boundary-obstructed TSC with MCMs (), and finally to a first-order TSC with chiral Majorana edge modes (h> h_{c2}). The topological nature of the first-order TSC with chiral Majorana edge modes is characterized by a nonzero Chern number, denoted as C, which corresponds to the total number of gapless chiral edge modes [Fig.2(c)] [56, 62, 63].

When , the bulk is always gapped and in topologically trivial phase (C=0). The 1D edge state of edge-IV closes at h=h_{c1}=1.4, and reopens as h>h_{c1}. This indicates that the 1D gapped normal edge state undergoes a topological phase transition in the presence of s-wave pairing. In this case, the 1D gapped normal edge state can be seen as a 1D nanowire. The topology of this class in one dimension can be characterized by a Z_2 number, which is associated with the Berry phase [64–70]. In our system, we calculate the Berry phase \gamma for the lower half bands to discriminate between topologically trivial and nontrivial phases. The Berry phase of each edge in our system can be obtained as follows. Without loss of generality, we consider OBCs in the \hat{x} direction and PBCs in the \hat{y} direction (x-OBC/y-PBC). In the parameter regions shown in Fig.2, the 1D edge state of edge-IV undergoes a closing-and-reopening transition in the energy gap, while the edge state of edge-III is always gapped. Therefore, the Berry phase \gamma of edge-IV can be calculated in the 1D Brillouin zone, k\in [ 0,2\pi ], with momentum k taking the values k_1, k_2, \cdots, k_N [64, 66],

(3)\gamma =-\mathrm{i}{\displaystyle \sum _l\log U(k_l),}

where U(k_l)={\phi }^{\ast } (k_l)\phi (k_{l+1}) is the link variable and \phi (k_l) are the eigenstates at momentum k_l. For multi-bands system, the link variable becomes U(k_l)=det\mathit{U}(k_l) with

(4)U_{i,j}(k_l)= {\phi }_i^{\ast }(k_l){\phi }_j(k_{l+1}),1\le i,j\le n ,

where {\phi }_i(k_l) are the eigenstate of the ith band, and n represents the filled bands. Similarly, we can also obtain the Berry phase of other edges. The mechanism here to obtain MCMs is based on gapped normal edge states, which is different from that based on gapless helical edge states whose topology is protected by bulk states [40, 71].

The numerical results depicted in Fig.2(d) illustrate the Berry phase of the current system. It is clearly shown that a finite Berry phase \gamma = \pi in Fig.2(d) corresponds to the presence of MCMs for . In this scenario, only the edge-IV resides in the topological superconducting phase, whereas the other three edges and bulk states are topologically trivial. Therefore, the whole system is a second-order TSC because the dimension of the MCMs is lower than the bulk by two, we term it a boundary-obstructed TSC. To summarize, we observe a transition from a second- to first-order topological superconducting phase as the Zeeman field increases, with the Berry phase and Chern number serving as distinguishing characteristics, respectively.

To verify the existence of Majorana bound states and uncover the topological nature of the phase transition, we present the spatial distribution of topological boundary states in real space. In Fig.3(a) and (b), h=1.5, we numerically diagonalize the Hamiltonian for a finite size (N_x=N_y= 40) using the same parameters as Fig.2. The corresponding low-energy spectrum exhibits a distinct gap and a pair of MZMs. It is found that the MZMs are localized at the intersection of topologically nontrivial and trivial edges, known as MCMs. This calculation also demonstrates that the boundary-obstructed TSC is protected by the topology of edge states rather than bulk states. In Fig.3(c) and (d), h=1.8, the system is into a first-order TSC. We observe the presence of gapless edge modes within the bulk energy gap. These chiral Majorana edge states extend along the edges of the system, aligning perfectly with the topological characteristics depicted in Fig.2.

In our modified structure, we can also realize the MCMs in other edges by tuning {\delta }_{1,2,3,4}. As depicted in Fig.4(a), {\delta }_1= 0.3 and {\delta }_{2,3,4}=0, the 1D edge state appears along edge-I, which can be induced into a topological superconducting phase. In this parameter region, the other three edges are topologically trivial, resulting in the localization of two MCMs at the two ends of edge-I. By tuning edge-III and edge-IV into a topologically nontrivial phase, four MCMs can exist at the four corners of the 2D lattice. Fig.4(b) illustrates the scenario where {\delta }_{1,3}=0.3 and {\delta }_{2,4}=0, and edge-II and edge-IV are topologically nontrivial. At the intersection of edge-II and edge-IV, there are no MCMs localized, as the two end Majoranas combine into an ordinary finite-energy fermion [11]. However, two MCMs can still be localized at the other two ends, as depicted in Fig.4(b). This demonstrates that the localization of MCMs can be controlled by tuning {\delta }_{1,2,3,4}. Besides, the topological phase transition at each edge can also be achieved by varying other accessible parameters such as the chemical potential or the Zeeman field. We can also employ spatially non-uniform chemical potentials and/or Zeeman fields to manipulate the local edge topology and then exchange the position of different MCMs [66, 72]. Hence, our modified model would be a promising platform to transfer and braid MCMs along the edges.

3.2 The generalized Majorana polarization

The MP characterizes the expectation value of the particle-hole operator and describes locally the Majorana nature. It has been demonstrated as a useful topological invariant to verify the existence of MZMs in 1D and quasi-1D TSCs [66, 67, 73–77], and even distinguish Majorana and trivial zero-modes [78]. In this subsequent discussion, we will generalize the MP as a useful topological invariant in detecting the existence of MCMs and chiral Majorana edge states in 2D lattices.

We start by defining a local MP at each site j as

(5)⟨{\mathrm{\Psi }}_j|{\mathcal{C}}_j|{\mathrm{\Psi }}_j⟩=-2{\displaystyle \sum _{\sigma }\sigma {\mu }_{j\sigma }{\nu }_{j\sigma }.}

In the Nambu basis, the particle-hole operator of a spinful system is \mathcal{C}={\mathrm{e}}^{\mathrm{i}\zeta }{\sigma }^y{\tau }^y\hat{\mathit{K}}, where \zeta is an arbitrary phase, {\tau }^y ({\sigma }^y) is the Pauli matrices in the particle-hole (spin) subspace, and \hat{\mathit{K}} is a complex-conjugation operator. {\mathrm{\Psi }}_j^T =({\mu }_{j\uparrow },{\mu }_{j\downarrow },{\nu }_{j\downarrow },{\nu }_{j\uparrow }) represent the eigenstate at site j, where \mu and \nu denote the electron and hole components, respectively. For a Majorana state, the local MP is fully aligned, referred to as “ferromagnetic” structure. The total MP of a Majorana state can be obtained by summing over the local MP. Additionally, within region R, where a Majorana state is localized, it must satisfy [66, 67, 73–77]

(6)P_M=\frac{|{\displaystyle \sum _{j\in R}⟨{\mathrm{\Psi }}_j|{\mathcal{C}}_j|{\mathrm{\Psi }}_j⟩|}}{{\displaystyle \sum _{j\in R}⟨{\mathrm{\Psi }}_j|{\mathrm{\Psi }}_j⟩}}=1.

In the following, we will demonstrate that the MP can also capture the topology of MCMs and chiral Majorana edge states. The MCMs are localized at the four corners of a 2D lattice, we therefore can simply set R=N_x/2\times N_y/2 in Eq. (6) with N_x(N_y) representing the length (width) of the 2D lattices. Fig.5(c) and (d) demonstrate the MP with a “ferromagnetic” structure in the MCMs. The numerical results illustrated in Fig.5(a) and (b) demonstrate that the phase diagram obtained by MP is consistent with Fig.2. As , the system is topologically trivial and the MP is therefore zero, P_M=0. In the region of , the system exhibits MCMs separated from the rest of the spectrum by a minigap, and P_M=1. When h>1.73, chiral Majorana edge states appear along the edges. The MP of chiral Majorana edge states shows locally “ferromagnetic” structure, but its direction changes slowly from site to site. Therefore, by summing over the local MP, we will get a nonzero but finite MP, P_M\simeq 0.8. We confirm that the MP is a reliable topological invariant in characterizing the successively transitions from a trivial phase to a boundary-obstructed TSC with MCMs, and finally to a first-order TSC with chiral Majorana edge modes. We establish a clear correspondence between MP and MCMs as well as chiral Majorana edge states. Here, the concept of MP is a generalization of that in 1D and quasi-1D TSCs. The direction of the MP is always opposite at the two ends of 1D nanowire or quasi-1D ribbon [73]. But the direction of MP for the two MCMs could not be opposite [Fig.5(d)]. Furthermore, the nonzero but finite MP can also capture the topology of chiral Majorana edge states, distinguishing it from MCMs and trivial states.

3.3 Topological phase diagram

To investigate the universality of our proposal and explore a wide range of topological behaviors, we will analyze the topological phase diagram as a function of chemical potential \mu and Zeeman field h. Firstly, let us examine the topology of the bulk states, which can be characterized by the Chern number. Refer to Fig.6(a) for better visualization. The color coding in the image represents different Chern numbers: dark blue for C=-1, light blue for C=0, yellow for C=1, and red for C=2. For the case of C=0, we observe four distinct regions: A, B, C, and D. However, their topological properties are different. In region A, the system is trivial and exhibits a gapped behavior. In region B, we find two 1D edge states crossing at k_y= 0 with opposite Chern numbers. Similarly, in region C, two 1D edge states with opposite Chern numbers cross at k_x= \pi. Lastly, region D shows two 1D edge states with opposite Chern numbers crossing at k_x= 0 and k_x=\pi, respectively. The energy gap \mathrm{\Delta }{E}_x along the \hat{x} direction [Fig.6(b)] and \mathrm{\Delta }{E}_y along the \hat{y} direction [Fig.6(c)] further confirm our observations.

In Fig.6(c), the dark blue region represents a gapless scenario along the \hat{y} direction. It is interesting to note the presence of two “islands” in the dark blue region. The small “islands” corresponds to the region D mentioned earlier in Fig.6(a), where the energy band is gapped along \hat{y} direction, but two 1D edge states exist along \hat{x} direction. In this region, there are no MCMs under OBCs even when the Berry phase \gamma =\pi. This demonstrates the competition between the topology of edge and bulk states. When the bulk states exhibit topologically protected edge states, the boundary-obstructed topological phase is not protected by the topology of the boundary. On the other hand, the bigger “island” signifies a boundary-obstructed TSC with MCMs, where the bulk states exhibit topologically trivial behavior.

Fig.6(d1)−(d3) demonstrate the successive transitions from a trivial phase to a boundary-obstructed TSC with MCMs, and eventually to a first-order TSC with chiral Majorana edge modes as the chemical potential \mu increases (along the dotted line in Fig.6(c)). The sequence of transitions is similar to those induced by the Zeeman field, except for \mu > 1.35. When \mu >1.35, the bulk states of the system enter a first-order phase characterized by C=-1. No MCMs exist under OBCs even when the Berry phase is still \gamma =\pi. It again demonstrates that the topological boundary states is first determined by the topology of bulk states, and then by the topology of system boundary when the bulk states are trivial.

In Fig.6(e) and (f), we explore the stability of the MCMs and MP against random disorders. For concreteness, we investigate two types of disorder: random disorder on the chemical potentials \mu +\delta \mu (\delta \mu =[-W,W] ) and random disorder on Zeeman field h+\delta h (\delta h= [-W,W]), where W represents the strength of the disorder. A 2D finite system with an OBC along \hat{x} and \hat{y} directions (N_x= N_y=40) is considered. The energy gap decreases but the MCMs always exist at larger disorder. It is evident that the MCMs and MP exhibit remarkable stability against disorders over a wide range of disorder strengths. Our calculation demonstrates that the MCMs are protected by the topology and robustness against different types of disorder.

3.4 Hinge Majorana zero modes

The results obtained on the 2D square lattice can be extended to three dimensions. We construct a 3D structure, denoted as H_{3\mathrm{D}}, by stacking our 2D model, H_{2\mathrm{D}}. The Hamiltonian is described by

(7)H_{3\mathrm{D}}={\displaystyle \sum _z{H}_{2\mathrm{D}}+{\displaystyle \sum _{x,y,z,\alpha ,\sigma }{t}_z{\alpha }_{x,y,z,\sigma }^{†}{\alpha }_{x,y,z\pm 1,\sigma }.}}

t_z represents the interlayer coupling between different 2D layers, and \sigma =\uparrow ,\downarrow denotes the spin, while \alpha =a,b,c ,d sublattices. Here, we assume t_z=1, but similar topological behaviors can be obtained for other values of t_z. The other parameters of H_{2\mathrm{D}} remain the same as Fig.3, representing a 2D boundary-obstructed TSC with MCMs.

To analyze the system, let us begin with a square geometry with lengths N_x=N_y= 40 in the x−y plane for the tight-binding model, but keep the Bloch momentum k_z along z well-defined. The band structure of the tight-binding Hamiltonian is plotted in Fig.7(a), where we can observe a zero-energy flat band corresponding to the hinge MZMs [79]. The corresponding zero-energy wavefunction is illustrated in Fig.7(b). The inset of Fig.7(b) displays the energy spectrum under OBCs along \hat{x}, \hat{y} and \hat{z} directions, revealing the emergence of hinge MZMs. When the 2D layer H_{2\mathrm{D}} is in a trivial phase, the stacking 3D structure is also in a trivial phase without hinge MZMs. Furthermore, the hinge MZMs are robustness against disorder. Our proposal offers an effective method to achieve a higher-order TSC in a 3D structure by stacking a 2D boundary-obstructed TSC.

4 Conclusions

In summary, we propose a method for designing first- and second-order TSCs on a 2D square lattice by controlling the topology of the edge and bulk states. This system can be realized in an optical lattice, where the SOC can be generated using spatially varying laser fields [80–83], and s-wave pairing can be achieved through the proximity effect or in a s-wave superfluid of ultracold fermionic atoms [56]. Here are the steps to transition from a second- to a first-order TSC: Firstly, we induce the emergence of a 1D gapped normal edge state by introducing a flux of \pi through each supercell in the 2D lattice. Secondly, using these 1D gapped edge states, we construct a boundary-obstructed TSC where the bulk states are topologically trivial. It is noteworthy that the MCMs are localized at the intersection of 1D nontrivial/trivial edges, and their topological properties are protected by the Z_2 topological invariant known as the Berry phase. By manipulating the topology of different 1D edge states, we can control the localization of the MCMs. Thirdly, by further increasing the Zeeman field, we observe a transition from a second- to a first-order TSCs by closing the bulk gap. The topology of this transition is protected by the bulk states and can be characterized by a nonzero Chern number. Moreover, in the region where the first-order and boundary-obstructed topological phases coexist, no MCMs are present, and the topological boundary states are determined by the topology of the bulk states. Although we have focused on the results for the square lattice, we expect these different order TSCs to also arise in other lattices with controllable topology between edge and bulk states.

Furthermore, we extend the concept of MP as a topological invariant to verify the existence of MCMs and chiral Majorana edge states in 2D TSCs. The MP is widely used to verify the existence of Majorana end states in 1D and quasi-1D TSCs previously. The derived phase diagram aligns well with the band gap, Berry phase and Chern number. The MP is a useful topological invariant for characterizing Majorana modes, even in complicated structures where finding an appropriate invariant is challenging. Additionally, we propose an effective method for achieving higher-order TSCs in a 3D structure by stacking a 2D boundary-obstructed TSC. This work deepens our understanding on TSCs and facilitates the study of the interplay between different orders of topology.

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