Tunable Majorana corner modes in noncentrosymmetric superconductors: Tunneling spectroscopy and edge imperfections

Majorana corner modes appearing in two-dimensional second-order topological superconductors have great potential applications for fault-tolerant topological quantum computations. We demonstrate that in the presence of an in-plane magentic field two-dimensional ($s+p$)-wave superconductors host Majorana corner modes, whose location can be manipulated by the direction of the magnetic field. In addition, we discuss the effects of edge imperfections on the Majorana corner modes. We describe how different edge shapes and edge disorder affect the number and controllability of the Majorana corner modes, which is of relevance for the implementation of topological quantum computations. We also discuss tunneling spectroscopy in the presence of the Majorana corner modes, where a lead-wire is attached to the corner of the noncentrosymmetric superconductor. The zero-bias differential conductance shows a distinct periodicity with respect to the direction of the magnetic field, which demonstrates the excellent controllability of the Majorana corner modes in this setup. Our results lay down the theoretical groundwork for observing and tuning Majoran corner modes in experiments on ($s+p$)-wave superconductors.

Moreover, while most previous studies on tunable MCMs consider square superconducting islands with clean edges, experimental implementations will most certainly deviate from such an ideal geometry. Thus, as the second objective of this paper, we study the effects of edge imperfections on the MCMs. For a square geometry with clean edges, we can find two stable Majorana zero modes. However, for more complicated edge configurations, the number of MCMs becomes more than two and depends on the applied magnetic field direction. In this case, we can no longer find a stable pair of Majorana zero modes that can make a full circle around the system, which complicates the braiding process. Thus, the edge configuration is an important factor for accomplishing the topological quantum computations in experiments.
At present, to the best of our knowledge, there are no specific theoretical proposals for detecting the tunable MCMs. Thus, as the third objective of this paper, we study the tunneling spectroscopy in the presence of the MCMs, where the normal lead wire is attached to the vicinity of the corner of the noncentrosymmetric superconductor. We demonstrate that the zero-bias conductance becomes a periodic function with respect to the di-rection of the applied magnetic field. This characteristic periodicity is understood by the fact that the zero-bias conductance is enhanced only when the MCM exits at the corner connected with the lead wire. As a result, we propose a smoking-gun experiment for the detection of controllable MCMs.
The organization of this paper is as follows. In Sec. II, we first derive an effective edge theory describing the emergence of tunable MCMs in a noncentrosymmetric superconductor with in-plane magnetic fields. Then, we confirm the validity of the edge theory by employing numerical simulations on a two-dimensional tight-binding model. In Sec. III, we study the effects of edge imperfections on the tunable MCMs. Specifically, we calculate the local density of states (LDOS) for evaluating the stability and controllability of MCMs in non-square superconducting islands. In Sec. IV, the differential conductance of normal lead wire/noncentrosymmetric superconductor hybrids is calculated by using lattice Green's function techniques. We demonstrate that the zero-bias conductance shows the periodicity as a function of the direction of the applied magnetic field, which serves as a fingerprint of the controllable MCMs. Our conclusion is given in Sec. V.

A. Model
Let us consider a two-dimensional noncentrosymmetric (s+p)-wave superconductor in the presence of an in-plane magnetic field. We start with a Bogoliubov-de Gennes (BdG) Hamiltonian in momentum space [66][67][68], where m is the effective mass of an electron, µ denotes the chemical potential, k F = √ 2mµ/ represents the Fermi wave number, and k = k 2 x + k 2 y . The strength of Rashba spin-orbit coupling is given by λ with g(k) = (k y , −k x , 0). The pair potential∆(k) contains both a spin-singlet s-wave component ∆ s and a spin-triplet p-wave component d(k) satisfying d(k) g(k) [66]. The Zeeman potential induced by the externally applied in-plane magnetic field is described by V Z = (V Z cos θ Z , V Z sin θ Z , 0), with θ Z representing the angle of the magnetic field measured from the x direction. In what follows, without loss of generality, we assume λ, ∆ s , ∆ t , V Z ≥ 0. The Pauli matrices in spin space are given byσ = (σ x ,σ y ,σ z ), and the 2 × 2 unit matrix is denoted by σ 0 .
We briefly discuss the topological property of the pure noncentrosymmetric (s + p)-wave superconductor described byȞ ncs (k). The positive eigen energies of H ncs (k) are given by with η = ±. While the spectrum of E ncs + (k) has a finite superconducting gap irrespective of the parameters, the superconducting gap in E ncs − (k) vanishes when For ∆ s > ∆ c (∆ s < ∆ c ), the BdG HamiltonianȞ ncs (k) can be deformed into the BdG Hamiltonian of a pure spin-singlet s-wave (pure spin-triplet helical p-wave) superconductor without any gap closing by decreasing λ and ∆ t (∆ s ) adiabatically. The spin-triplet helical pwave superconductor is well known as a time-reversal invariant topological superconductor characterized by a Z 2 topological invariant [74,75], wheres the spin-singlet superconductor is topologically trivial. Since the topological invariant never changes without gap closing, the noncentrosymmetric (s + p)-wave superconductor with ∆ s < ∆ c is topologically equivalent to the spin-triplet helical p-wave superconductor and host the topologically protected helical Majorana edge states [57][58][59][60][61]. In what follows, we only focus on the noncentrosymmetric (s + p)-wave superconductor in the topologically nontrivial phase (i.e., ∆ s < ∆ c ). Then, we discuss the effects of the Zeeman potential against the bulk energy spectrum. Within first-order perturbation theory with respect to V Z , the energy spectrum is given by with θ k = arctan(k y /k x ). Although the Zeeman potential causes a non-monotonic energy shift, it is clear that the Zeeman potential suppresses the superconducting gap size, and a large enough Zeeman potential may bring the system into a gapless superconducting state similar to the case of the pure helical p-wave superconductor discussed in Ref. [76]. Thus, throughout this paper, we only assume weak enough Zeeman potentials, such that the bulk superconducting gap remains finite and large enough. Although the superconducting gap remains finite, the noncentrosymmetric (s + p)-wave superconductor is no longer characterized by the Z 2 topological invariant due to broken time-reversal symmetry. Nevertheless, in the following, we demonstrate that the noncentrosymmetric (s + p)-wave superconductor under the in-plane magnetic field becomes a second-order topological superconductor hosting MCMs.

B. Effective Edge Hamiltonian
In order to discuss the emergence of MCSs in the present system intuitively, we derive an edge theory, similar to the ones discussed in Refs. [41, 43, and 49]. We here assume that the present superconductor has an edge perpendicular to n = (cos γ, sin γ, 0), where the spatial coordinate along the direction perpendicular (parallel) to the edge is represented by x ⊥ (x ), and the superconductor occupies the entire half-space x ⊥ ≤ 0 (see Fig. 1). In addition, we apply periodic boundary conditions in the direction parallel to the edge (i.e., the direction along the x -axis). To obtain an effective edge Hamiltonian, we rewrite the BdG Hamiltonian in momentum space as, with where k ⊥ (k ) represent the momentum perpendicular (parallel) to the edge of the superconductor. In what follows, we treatȞ (k) andȞ Z as the perturbations. This approximation is justified when k /k F ≪ 1 and when V Z is significantly smaller than the superconducting gap. Moreover, in the following analysis, we assume ∆ s , ∆ t ≪ λk F , µ. For the zeroth-order perturbation, we replace k ⊥ → −i∂ x ⊥ and find zero-energy states satisfying the equation,Ȟ with the boundary condition ψ(0) = ψ(−∞) = 0. As a result, we find the two solutions, with ζ = ±, where we neglect the insignificant terms of . The matrix elements of the perturbation termsȞ (k) +Ȟ Z within the first order are calculated as As a result, we obtain an effective edge Hamiltonian, which is unitarily equivalent to  (31), we clearly find that the mass term for the linearly dispersive helical edge states is given by V = V Z sin(θ Z − γ). Importantly, the sign of the mass term is determined only by the relative angle between the Zeeman field V Z and the edge normal vector n: Thus, there is a possibility that two adjacent edges of the system have the mass terms with opposite signs. In such case, we obtain zero energy states bounded in the vicinity of the corner (i.e., MCMs) because the mass term has a kink there [41,43,49]. In the next section, we confirm the validity of our edge theory by performing numerical calculations of a tight-binding model.

C. Tunable Majorana corner modes
We here consider a square superconducting island as shown in Fig. 2(a), where the four corners are labeled as A, B, C, and D, respectively. When we apply the Zeeman field along θ Z = 0.25π, according to Eq. (32), the edges AB and DA have negative mass terms, whereas the edges BC and CD have positive mass terms. Therefore, based on the effective edge theory, we can expect MCMs at the corners B and D. To confirm the above statement, we numerically calculated the LDOS by using the formula ρ S (r, E) = −Tr Im Ǧ (r, r, E + iδ) /π, whereǦ(r, r ′ , E) represents the Green's function and Tr denotes the trace in spin and Nambu spaces. δ is a small imaginary part added to the energy E. The Green's function is calculated on a tight-binding model by use of a recursive Green's function technique [77], where we replace ξ(k) → 2t(1−cos k x )+2t(1−cos k y )−µ, g(k) → (sin k y , − sin k x , 0), and ∆ t /k F → ∆ t . The explicit expression of the BdG tight-binding Hamiltonian is given in Appendix A. In Fig. 2(a), we show the LDOS at zero-energy for a (201 × 201) square island. We use the parameters: µ = t, λ = 0.5t, ∆ t = 0.5t, ∆ s = 0.2∆ t , V Z = 0.5∆ t , and δ = 10 −4 ∆ t . The LDOS is normalized by ρ N = ρ N (r, E = 0) , where ρ N (r, E) representing the LDOS for the normal state (i.e., ∆ t = ∆ s = 0), and · · · means the averaged value with respect to the lattice sites. Moreover, we plot log 10 (ρ S /ρ N ) instead of the raw data of ρ S . As shown in Fig. 2(a), the LDOS at zero energy has a significant amplitude only in the vicinity of the the corners B and D. This result confirms the presence of MCMs, as expected. In Fig. 2(b), we show the density of states (DOS) as a function of the energy. The DOS is calculated by D S (E) = r∈S ρ S (r, E), where r∈S represents the summation over the lattice sites in the superconductor island. The results are normalized by the DOS in the normal state, D N = (201 × 201)ρ N . As shown in Fig. 2(b), the DOS exhibits a sharp zero energy peak inside the superconducting gap ∆ eff = 0.31∆ t . Due to particle-hole symmetry of the superconductor, the two MCMs can depart from zero energy only when they hybridize with each other, thereby becoming conventional electron-like and hole-like quasiparticle modes. Therefore, as long as the two MCMs are spatially separated and protected by the superconducting gap ∆ eff , they can remain robust at zero energy.
Next, we discuss the controllability of the MCMs. In Fig. 3, we show the LDOS at zero energy with various magnetic field directions, namely (a) θ Z = 0.375π, (b) 0.5π, (c) 0.625π/8, and (d) 0.75π. When θ Z = 0.375π, according to Eq. (32), the kinks of the mass term are located at the corners B and D. As a result, although the spatial distributions of the MCMs are slightly modulated from the case with θ Z = 0.25π, we can still find them at the corners B and D. When θ Z = 0.5π, the mass term becomes zero at the entire edges AB and CD. As a consequence, zero-energy Majorana edge states appear on the entire edges AB and CD. The emergence of Majorana edge states in a time-reversal invariant topological superconductor under a Zeeman field with a certain direction has been discussed in terms of the Majorana Ising spin [59,[78][79][80], and it has been shown that the energy spectrum of the edge states becomes gapless (strictly speaking, the energy spectrum is discrete due to the finite size effect) [41,59,[78][79][80]. When θ Z exceeds 0.5π, the positions of the mass term kink move to the corner A and C. As a consequence, for θ Z = 0.625π and 0.75π, the MCMs appear at the corners A and C. By varying the direction of the Zeeman field from θ Z = 0.25π to 0.75π, the MCM originally located at the corner B hop to the corner A. In the same way, the pair of the MCMs can make a full circle around the system by rotating the direction of the Zeeman field by 2π.

III. EFFECT OF EDGE IMPERFECTIONS
In the above, we have demonstrated that the noncentrosymmetric (s + p)-wave superconductor can host tunable MCMs. However, in real experiments, the superconducting island generally deviates form the perfect square shape. Thus, in this section we study the effects of different island shapes (subsection III A) and of edge disorder (subsection III B). As an example of a non-square shape, we consider a square island, where a square regions of size (L c × L c ) has been removed from the upper right corner B [41], as shown in Fig. 4. We label the six corners as A, B I , B II , B III , C, and D. We first consider the MCMs for a field applied along the direction θ Z = 0.25π. Under this circumstance, according to Eq. (32), the mass term changes its sign at the four corners B I , B II , B III , and D. In Fig. 5, we show the LDOS at zero energy for an island with the shape of Fig. 4 with (a) L c = 60 and (b) L c = 15. In accordance with the edge theory, as shown in Fig. 5(a), we find four MCMs for the island with L c = 60. However, for an island with L c = 15, there are only two MCMs, where one of them is spread across the corners B I , B II , and B III . This implies that the three MCMs originally located at B I , B II and B III hybridize with each other thereby becoming non-degenerate, with two states at small non-zero energy and one state at zero energy, in accordance with particle-hole symmetry. In fact, the edge theory is valid only when the edges are sufficiently longer than the decay length of the MCMs because, to derive the edge theory, we apply the periodic boundary condition in the direction parallel to the edge and do not take into account the length of the edge. Roughly speaking, the decay length of the MCM is evaluated by the inverse of the superconducting gap [81], i.e., ξ eff ≈ t/∆ eff , where ξ eff = 6.5 with the present parameter choices. Thus, when L c exceeds ξ eff significantly, we obtain the MCMs at the three corners B I , B II and B III . In Fig. 5(c), we show the DOS at zero energy as a function of L c , where the DOS is normalized by D sq (E = 0) representing the zero-energy DOS for the ideal square island, calculated in Fig. 2(b). When L c is large enough (i.e., L c 30), we find D S /D sq = 2. Although the DOS does not count directly the number of the states, the relation of D S /D sq = 2 implies that the number of the MCMs in the present island is twice as much as that in the ideal square island: there are four MCMs as shown in Fig. 4(a). For L c 16, we find D S /D sq ≈ 1 suggesting the presence of two MCMs as shown in Fig. 4(b). In the intermediate region (i.e. 16 L c 30), the DOS shows non-monotonic oscillations, which originate from interference effects between the three MCMs, which modify their mutual couplings.
In Fig. 6(a), we show the DOS at zero energy as a function of the angle of the Zeeman field, where we vary 0 ≤ θ Z ≤ 0.625π and choose L c = 60. We find D S /D sq ≈ 2 for 0.1π θ Z 0.4π, whereas D S /D sq ≈ 1 for 0 θ Z 0.06π and 0.45π θ Z . Therefore, the number of the MCMs changes by rotating the Zeeman potential. To see more details, in Figs. 6(b)-6(i), we present the LDOS at zero energy for eight different values of θ Z . When θ Z = 0, as shown in Fig. 6(b), we find the Majorana edge states at the edge B III C, whereas there are no significant enhancement in the LDOS at the edge B I B II . When θ Z = 0.03π, as shown in Fig. 6(c), an MCM appears at the corner B III , where the wave function is extended towards the corner C. In addition, we find the small enhancement of the LDOS at the edge B I B II . Strictly speaking, this small enhancement is not originated from exact zero energy states but from the finite energy states having energy extremely close to zero. We cannot eliminate contributions from such low-energy states because, although the strict definition of the LDOS is given by ρ S (r, E) = − lim δ→0+ Tr Im Ǧ (r, r, E + iδ) /π, the small imaginary part of δ is always set to be finite in our numerical calculations. As shown in Fig. 6(d), the LDOS at the edge B I B II increases by increasing θ Z . When θ Z = 0.1π, as shown in Fig. 6(e), we find that the zeroenergy LDOS is significantly enhanced in the vicinity of the corners B I and B II . As already shown in Fig. 5(a), with θ Z = 0.25π, we obtain three distinct MCMs at the three corners B I , B II , and B III . As a consequence, by increasing θ Z from zero to 0.25π, two additional MCMs are created at the corners B I and B II , while a single MCM stays constantly at the corner B II . When θ Z = 0.4π, as shown in Fig. 6(e), the wave function of the MCM at the corner B III is extended towards the corner B II , such that the MCMs at the corners B II and B III start to overlap with each other. When θ Z = 0.45π, as shown in Fig. 6(e), the LDOS at the edge B II B III is strongly suppressed. With θ Z = 0.5π, we can no longer find the enhancement of the LDOS at the edge B II B III , whereas we find a single Majorana edge state at the edge AB I . With θ Z = 0.625π, we find only two MCMs at the corner A and C. Therefore, by increasing θ Z from 0.25π to 0.625π, the MCMs at the corner B II and that at B III start to hybridize with each other and move away from zero energy. At the same time the single MCM originally located at the corner B I hops to the corner A, while the single MCM of corner D hops to corner C. Importantly, in this process, the MCMs originally appearing at B II and B III vanish by hybridizing with each other, and never reappear at any other corner. To accomplish the braiding process in topological quantum computations, we must exchange the positions of the two Majorana zero modes forming a pair. In the present edge configuration, however, we can no longer find a stable pair of MCMs that can fully circle the edges of the superconductor, and therefore fail the braiding process. To avoid the emergence of additional and undesired MCMs, as also shown in Fig. 5, we have to eliminate additional corners whose adjacent edges are longer that the decay length of the MCMs.

B. Edge roughness
Next, we consider a square island with rough edges. To described the rough edges, we remove the outermost sites from an ideal square island with probability p, and repeat this etching process X times [82,83]. With this process, there is a possibility that small debris separated from the largest island are created. Thus, we also remove such debris. In Figs. 7(a) and 7(b), we show the LDOS at zero energy and the DOS as a function of the energy, respectively. We apply the Zeeman field along θ Z = 0.25π and consider a rough edge with (p, X) = (0. 1,20). We find that the zero-energy LDOS in the vicinity of the corners B and D is enhanced significantly. In addition, as shown in Fig. 7(b), the DOS retains a steep zero energy peak structure, whereas we also find finite DOS inside the effective superconducting gap in the clean limit ∆ eff = 0.31∆ t . As also discussed in Sec. II C, the MCMs are protected by the particle-hole symmetry as long as they are sufficiently separated spatially. Thus, the MCMs remain robust even in the presence of weak or moderate edge roughness that does not destroy the superconducting gap completely.
In Fig. 8(a), we show the normalize DOS at zero energy as a function of the direction of the Zeeman field. For the solid red line, we use the same superconducting island as in Fig. 7(a), i.e., (p, X) = (0.1, 20). For the dotted black line, we consider an island with weaker roughness (p, X) = (0.1, 5). At first, we concentrate on the results with (p, X) = (0.1, 20), shown by the solid-red line. For 0.1π θ Z 0.4π and for 0.69π θ Z 0.89π, we find D S /D sq ≈ 1 suggesting the presence of distinct MCMs. However, in the vicinity of θ Z = 0, 0.5π, and π, the DOS shows a non-monotonic dependence and becomes clearly larger than one, i.e., D S /D sq > 1. To see more details, in Fig. 8(b), we show the LDOS at zero-energy with θ Z = 0.47π, where we use the same superconducting island for calculating the solid-red line in Fig. 8(a). For comparison, in Fig. 8(c), we also show the result with the clean edge. With the dirty edge, we find that the LDOS is enhanced not only in the vicinity of the corner B and D but also around the center of the edge AB and at the corner C. According to the effective edge theory, when θ Z is close to 0.5π, the mass term V = V Z sin(θ Z − γ), which is the source of the energy gap in the edge spectrum, is suppressed at the edges AB and CD. Particularly, the mass term of the edges AB and CD vanishes completely at θ = 0.5π. The results in Fig. 8 suggest that the already small energy gap is reduced further by the edge roughness, such that states with extremely small (but finite) energy appear at the edges AB and CD. In the topological quantum computations, the energy gap which energetically separates the MCMs from other finite energy states plays an essential role to suppress the decoherence during the braiding operations. Therefore, the edge roughness causing damage to the energy gap in the edge spectrum affects negatively the braiding processes of the MCMs. As shown by the dotted black line in Fig. 8(a), the undesirable enhance-  ment in the low-energy DOS is suppressed by decreasing the edge roughness.
In the above, we have shown that, although the MCMs exist robustly, the braiding operation of them may be disturbed by edge imperfections. The simplest countermeasure would be fabricating superconducting islands with clean and smooth edges. Even so, establishing a practical way to to make the braiding process more robust would be a desirable work [52][53][54].

IV. TUNNELING SPECTROSCOPY
Finally, we propose an experiment for detecting the tunable MCMs. Let us consider a normal lead wire/superconductor junction, where the normal lead wire is attached to a corner of the superconductor as shown in Fig. 9. We describe the present junction with a tight-binding model, where a lattice site is indicated by r = jx + my with x (y) representing the unit vector in the x (y) direction. The superconductor (lead wire) is located at 1 ≤ j ≤ ∞ (−∞ ≤ j < 1) and 1 ≤ m ≤ W S (1 ≤ m ≤ W N ), where W S (W N ) denotes the width of the superconductor (lead wire). The superconducting segment is described by the BdG Hamiltonian used also in the above numerical calculations, and the normal lead wire is described by setting ∆ t = ∆ s = 0 and λ = 0. The hopping integral between the superconduc-tor and the lead wire (i.e., j = 0 and j = 1) is given by t int . The explicit form of the BdG Hamiltonian describing the present junction is given in Appendix A. In the following calculations, we fix the parameters as µ = t, λ = 0.5t, ∆ t = 0.5t, ∆ s = 0.2∆ t , V Z = 0.5∆ t , t int = 0.1t, W N = 10, and W S = 101. We assume a sufficiently low transparency at the junction interface (i.e., t int = 0.1t) such that the bias voltage is mainly dropped at the interface. Under this circumstance, we can calculate the differential conductance at zero temperature by using the Blonder-Tinkham-Klapwijk formula [84][85][86], where r ee ζ,ζ ′ and r he ζ,ζ ′ denote a normal and Andreev reflection coefficient at energy E, respectively. The indexes ζ and ζ ′ label an outgoing and incoming channel in the normal lead wire, respectively. These reflection coefficients are calculated using recursive Green's function techniques [77,87].
In Fig. 10(a), we show the zero-bias conductance G(eV = 0) as a function of the angle of the Zeeman field θ Z . We find the conductance plateaus at 2e 2 /h for 0 < θ Z < 0.5π and π < θ Z < 1.5π. For these θ Z , according to the effective edge theory in Sec. II B, the MCM appears at the corner attached to the lead wire. Therefore, it is clear that the conductance plateaus is caused by the resonant tunneling through the MCM. To see more details, in Fig. 10(b)-10(g), we show the differential conductance as a function of the bias voltage for various θ Z . At θ Z = 0.25π, as shown in Fig. 10(b), we find a steep zero-bias conductance peak, where the conductance spectrum for finite energies displays an almost hard-gap structure. For θ Z = 0.375π, we still find the clear peak, whereas the superconducting gap edge has now a tail that was absent for θ Z = 0.25π. When θ Z = 0.5π, the Majorana edge mode appears along the edge perpendicular to the junction interface. Since the Majorana edge mode is not bounded along the junction interface, it cannot cause resonant tunneling [88]. As a result, as shown in Fig. 10(d), the zero-bias conductance peak disappears  when θ Z = 0.5π. At θ Z = 0.75π, there are no zeroenergy states in the vicinity of the corner attached to the lead wire. As a consequence, we find a hard gap in the conductance spectrum. When θ Z = π, the gapless Majorana edge states appear at the edge along the junction interface. Here, we must note that the energy levels of the Majorana edge states are discretized owing to the finite size of the system. As a consequence, as shown in Fig. 10(f), the conductance spectrum exhibits many spikes for low bias voltages. When we prepare a sufficiently large superconductor, the energy levels of the Majorana edge states will form a continuum, such that the conductance spectra show a broad peak structure instead of a collection of sharp peaks. With θ Z = 1.125π, the MCM again appears at the corner attached to the lead wire. As a consequence, as shown in Fig. 10(g), the conductance spectrum exhibits a zero bias peak, whereas we also find significant enhancement in low bias voltages. When θ Z = 1.25π, the conductance spectrum returns to that in Fig. 10(b).
The periodicity of the zero-bias conductance with respect to the direction of the Zeeman field is a smoking gun evidence for the existence of controllable MCMs. In real experiments, finite temperature effects and other perturbations may modify the details of the conductance spectrum. Moreover, the edge roughness may disturb the stepwise change in the zero-bias conductance shown in Fig. 10(a) because according to Sec. III B, the edge roughness affects significantly the spectrum of the edge states around θ = 0, 0.5π, π and 1.5π. Nonetheless, our proposal is still valid for identifying the controllable MCMs because we only need the distinct periodicity in the zero bias conductance, and we do not need any detailed and quantitative informations of the conductance spectrum.

V. CONCLUSION
We have demonstrated that a two-dimensional noncentrosymmetric (s+p)-wave superconductor in the presence of an in-plane magnetic field can host tunable Majorana corner modes (MCMs). We also show that the number of MCMs depends sensitively on the edge shape of the noncentrosymmetric superconductor. We find that with irregular edge shapes there are no stable pairs of MCMs that can fully circle around the edges of the superconductor, a property that is important for braiding. We have also studied the effects of edge roughness on the MCMs and found that the MCMs are in general robust to edge roughness. However, edge roughness suppresses the edge gap and may cause undesirable decoherence effects during the braiding process. As a result, although the MCMs can be found insensitive to the edge configurations, the edge imperfections may disturb the realization of topological computations. Therefore, developing practical measures to stabilize the braiding process of tunable MCMs, as well as fabricating clean devices, would be the important future tasks for realizing topo-logical quantum computations. In addition, we discuss the tunneling spectroscopy in the presence of the MCMs, where a normal lead wire is attached to a corner of the noncentrosymmetric superconductor. The periodicity in the zero-bias conductance with respect to the magnetic field direction is a smoking-gun signature for of the existence of controllable MCMs. The proposed experiment is promising for observing the tunable MCMs, because we can expect the periodicity in the zero-bias conductance to be insensitive to the details of the model. In conclusion, we have provided groundwork knowledge for designing specific experiments for detecting tunable MCMs, which is an essential step for realizing future topological quantum computations using higher-order topological superconductors.
where c † r,σ (c r,σ ) is the creation (annihilation) operator of an electron at r with spin σ =↑, ↓, t denotes the nearest-neighbor hopping integral, µ is the chemical potential. The strength of the Rashba spin-orbit coupling is represented by λ. The Zeeman potential due to the externally applied in-plane magnetic field is given by V Z = (V Z cos θ Z , V Z sin θ Z , 0). The spin-singlet s-wave and spin-triplet p-wave pair potentials are denoted with ∆ s and ∆ t , respectively. The Pauli matrices in spin space are given byσ = (σ x ,σ y ,σ z ). The normal lead wire in Fig. 9 is described by H N with replacing L S → −∞, W S → W N and λ → 0. The coupling between the lead wire and the superconductor is described by j=1,m,σ c 0,m,σ + c † 0,m,σ c 1,m,σ , where t int denotes the hopping integral at the junction interface.