Topological Edge States of a Majorana BBH Model

: We investigate a Majorana Benalcazar–Bernevig–Hughes (BBH) model showing the emer-gence of topological corner states. The model, consisting of a two-dimensional Su–Schrieffer–Heeger (SSH) system of Majorana fermions with π ﬂux, exhibits a non-trivial topological phase in the absence of Berry curvature, while the Berry connection leads to a non-trivial topology. Indeed, the system belongs to the class of second-order topological superconductors ( HOTSC 2 ), exhibiting corner Majorana states protected by C 4 symmetry and reﬂection symmetries. By calculating the 2D Zak phase, we derive the topological phase diagram of the system and demonstrate the bulk-edge correspondence. Finally, we analyze the ﬁnite size scaling behavior of the topological properties. Our results can serve to design new 2 D materials with non-zero Zak phase and robust edge states.


Introduction
The notion of higher order topological phases first appeared for insulating systems (HOTIs) [1][2][3][4]. Indeed, a second-order topological insulator is a d dimensional system with gapped d − 1 dimensional boundaries and d − 2 localized modes (corner states in two-dimensional systems). This new topological phase can be protected by a variety of crystalline symmetries, such as reflection symmetries and C 4 symmetry [1].
Recently, a similar physics has also been explored in the context of superconducting systems (HOTSCs) [5][6][7][8][9]. Superconductors with such novel topological properties have attracted increasing attention as they possess surface states that propagate along one-dimensional curves (hinges) or are localized at some points (corners) on the surface. In particular, m-dimensional Majorana corner states can be realized in d-dimensional superconductors, with m ≤ d − 2.
In close analogy with Ref. [2], we consider a Benalcazar-Bernevig-Hughes (BBH) model of Majorana fermions and will show that it is a suitable model to realize an HOTSC with robust corner Majorana zero modes. It has been demonstrated that the original fermionic 2D SSH model cannot support topological corner states [10,11] unless one considers a negative coupling per plaquette, as theoretically proposed [2] and experimentally demonstrated in [12][13][14]. On the other hand, in our model, the mapping of Majorana fermions onto complex fermion operators gives rise to a model of Kitaev chains coupled by a staggered pairing coupling, showing a higher order topological phase. Despite being characterized by a specific set of parameters, our model could be realized in photonics systems where a high efficiency to control parameters through gauge fields has been demonstrated [15,16]. Furthermore, our model could have several interesting and potential applications, in particular, in studying the braiding dynamics of Majorana fermions [17] for condensed matter platforms. Topological insulators are phases of matter characterized by topological edge states that propagate in a unidirectional manner that is robust to imperfections and disorder. They propose a concept that exploits topological effects in a unique way: the topological insulator laser. These are lasers whose lasing mode exhibits topologically protected transport without magnetic fields. The underlying topological properties lead to a highly efficient laser, robust to defects and disorder, with single-mode lasing even at very high gain values. The topological insulator laser alters current understanding of the interplay between disorder and lasing, and at the same time, opens exciting possibilities in topological physics, such as topologically protected transport in systems with gain. On the technological side, the topological insulator laser provides a route to arrays of semiconductor lasers that operate as one single-mode high-power laser coupled efficiently into an output port.
The paper is thus organized as follows. In Section 2, we present the model and the lattice geometry. In Section 3, we discuss the phase diagram and check the bulkedge correspondence. Section 4 is devoted to the analysis of the finite size scaling of the topological properties. In particular, we discuss the behavior of the gaps closing and the non-local fermion correlation functions. Conclusions are drawn in Section 5. Appendix A contains details on the calculation of non-local fermion correlations.

Model
We consider a two dimensional lattice model of Majorana fermions with staggered couplings w, v which define a plaquette with π flux, (see Figure 1, panel (a)) described by the Hamiltonian: where a m,l and b m,l are the Majorana modes associated to a complex fermion operator c m,l = (a m,l + ib m,l )/2, L and N are, respectively, the length and the width of the system, m and l are the lattice sites. The lattice representation of our model is reported in Figure 1, panel (a), where the green plaquette is the unit cell and the Majorana operators a and b of Equation (1) are represented by two circles of different colors. In panel (b) of the same figure, we show the topological fully dimerized limit with w = 0, v = 0, which generalizes the same well-known limit discussed for the SSH model in [18]. In this limit, the four Majorana modes a 11 , a 1,N , b L,1 and b L,N become corner zero-energy modes which decouple from the lattice. The full topological phase diagram of the model will be discussed in Section 3. Expressing the Majorana operators in terms of complex fermion operators, the Hamiltonian in Equation (1) results in N Kitaev chains of length L coupled only by pairing terms (see Figure 1, panel (c)): (tc † m,l c m+1,l + ∆c m,l c m+1,l + h.c.) with µ = w, t = ∆ = −v/2 and ∆ 1 : In the next section, we will discuss the analysis of the topological phase diagram of the model by evaluating the Zak phase and the energy spectrum. Indeed, in (c), the following identities hold: µ = w, t = ∆ = −v/2 and ∆ 1 is staggered, with −iw associated with continuous red lines and −iv associated to dashed red lines.

Topological Phase
By imposing periodic boundary conditions to the model in Equation (1), we can apply the Fourier transform. The translational invariant unit cell is given by the green plaquette in Figure 1 (panel (a)) and we obtain a 4 × 4 matrix in the momentum space, where: The operator in Equation (4) satisfies reflection symmetries with reflection operators m x , m y exchanging, respectively, x → −x, y → −y and, therefore, k x → −k x , k y → −k y : where m x = σ z ⊗ σ y and m y = σ y ⊗ I. Because of the simple structure, the lattice is also invariant under rotations by θ = π/2 up to a gauge transformation. The C 4 symmetry, describing the rotation, is associated to the operator R 4 exchanging (k x , k y ) → (k y , −k x ): where σ ± = (σ 1 ± iσ 2 )/2, σ 0 is the identity , σ i with i = 1 . . . 3 are the Pauli matrices and (R 4 ) 4 = −1. The Hamiltonian in Equation (4) also satisfies particle-hole P = IK, with I = σ 0 ⊗ σ 0 and K the complex conjugation, time reversal T = σ z ⊗ σ z K and chiral symmetries C = σ 3 ⊗ σ 3 : thus the model belongs to the trivial two-dimensional BDI class [19]. Indeed, the Berry curvature B = ∇ ∧ A vanishes everywhere in the Brillouin zone due to the coexistence of T and reflection symmetries. Here, A n,m = ψ n |i∂ k |ψ m and |ψ m is the Bloch wave function associated with the m band. However, the reflection symmetries of Equation (5) ensure fractional quantized components of the polarization: P is also known as a 2D Zak phase [20,21]. In particular, the 2D Zak phase takes the value P = (1/2, 1/2) in a non-trivial phase and P = (0, 0) in the trivial one. The presence of C 4 symmetry ensures that the model described is a second-order topological system with intrinsic Majorana corner states associated to the fractional quantized values of P, which cannot be removed by a change of boundary preserving the bulk [22]. Indeed, for systems with spatial symmetries, the simultaneous presence of standard symmetries (C, P, T ) and crystalline symmetries is essential to realize a second-order topology. In analogy with the Aharonov-Bohm (AB) effect for a magnetic field [23], one observes the presence of a quantized charge polarization in the absence of the Berry curvature; this phenomenon gives rise to the higher order topological superconductivity [5][6][7][8][9]22]. In Figure 2 panel (a), we report the topological phase diagram (PD) of our system, which falls in a non-trivial regime for |w| < |v|. Panels (b) and (c) of the same figure show the band structure along the path in k-space: M → X → Γ → M, with M = (π, π), X = (π, 0), Γ = (0, 0). The energy bands are doubly degenerate and invariant under the exchange of w → v: More precisely, the system shows a phase transition at the critical point w = v, as can be seen from the gap closing at Γ = (0, 0) point in panel (c) (w = v = 1). Panel (b) of Figure 2 shows the bands for the points (w = 2, v = 1) and (w = 1, v = 2), in the trivial and non-trivial region, respectively. Looking at the expression of the energy bands in Equation (9), the two points are indistinguishable from the bulk properties point of view and a check of the bulk-edge correspondence is needed to show the different behaviour of topological and trivial points. To this end, in Figure 3, a system of finite size, L = 100 and N = 50, is analyzed. Panel (a) of Figure 3 shows the lowest energy spectrum of the system following the horizontal red cut in the phase diagram of Figure 2a. Here, we observe the expected correspondence between the gap closing and the appearance of non-trivial states. Non-trivial topological corner states associated with 2D Zak phases are shown in panel (b) of Figure 3, where we fix the parameters to those corresponding to the red circle of panel

Topological Properties of Finite Size Systems
In the thermodynamic limit, when |w| < |v|, we have seen that the Majorana BBH model exhibits four Majorana corner modes and the ground state manifold is fourfold degenerate. In particular, since the model preserves the parity, the system is twofold degen-erate for every parity sector. If the size is finite, there is a weak coupling between Majorana corner states, which decreases as the system size increases. In Figure 4, we report a scheme of the three lowest energy levels for the two parity sectors for a finite size strip and set the parameters in the topological regime. To analyze the topological properties of the finite size system, we study the behavior of: (a) the gaps between ground states and first excited states indicated by δ 0 , and δ 1 , (b) the energy separation between the second and third excited states, indicated by E 0 and E 1 and (c) the energy separation between the ground states corresponding to the two different parities ( ). The subscripts 0, 1 represent the two parities. In the thermodynamic limit, fixing the parameters in a topological point of the phase diagram, the following asymptotic limits are expected: δ 0 → 0, δ 1 → 0, → 0, while E 0 and E 1 should remain finite. More precisely, the manifold gaps δ 0 , δ 1 and are expected to decay exponentially to zero, while increasing the system size. Actually performing the analysis of the manifold gaps and of while varying the size of the system is the best way to test the robustness of non-trivial regime.    (c)). , on the other hand, turns out to be comparable, for all the considered sizes, to the machine error. This behavior of the gaps ensures that the topological phase of the system is robust.
To gain further insight into the topological properties of the finite size system in Figure 6, we show the non-local fermion correlations, where panels (a) and (b) correspond to < c † 1,1 c m,l > and < c † L,N c m,l >, respectively, with lattice indices l and m running along the length and the rung of the strip with L = 100 and N = 30. The shorthand notation < c † 1,1 c m,l > and < c † L,N c m,l > instead of GS|c † 1,1 c m,l |GS and GS|c † L,N c m,l |GS is used, with |GS being the many body ground state. All the details about the formulas used to compute the fermion correlations are reported in Appendix A. As we can see, the fermionic states are always correlated along the shortest directions (1, 1) − (1, N) and (L, 1) − (L, N). The same behavior is observed when the dimensions of the strip are exchanged L → N, N → L. In panels (c) and (d) of Figure 6, we report the non-local fermion correlations of a Majorana BBH strip with L = 30 and N = 100. We observe the tendency of the system to form topological modes that correlate along the shortest dimension. A finite size square system L = N is thus the best configuration to observe the formation of Majorana fermions at the corners.

Conclusions
We have proposed and analyzed the topological phases of a Majorana BBH model, consisting of a Majorana 2D SSH model with a synthetic magnetic flux of π. Due to the coexistence of C 4 symmetry and reflection symmetries, the system turns out to be a second-order topological superconductor with a quantized 2D Zak phase in the absence of Berry curvature. Starting from the Hamiltonian in k-space, we have studied the topological phases of the system, showing the bulk-edge correspondence and the formation of Majorana corner states. To study the robustness of the topological phase, we have analyzed the finite size scaling of the gaps between the ground states and the first excited states in the two parity sectors and found the expected exponential decay with the system length. The analysis of the fermion non-local correlation functions shows that the Majorana zero modes nucleate at the corners of the system, correlating along the shortest dimension.
Our model could be realized in photonics systems or in cold atoms platforms where a high efficiency to control parameters through gauge fields has been demonstrated. We believe that our proposal opens new directions in inducing unique corner states by gauge field and offering possibilities to design 2D models with non-zero Zak phase and robust edge states. Various directions remain open, among which the study of the effect of interactions on the higher order topological phase is a relevant one.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.
The Γ is expressed in terms of the commutator between Majorana operators a i and b j , where for simplicity, we adopt the notation φ|[a, b]|φ =< [a, b] > introduced in the main text. Choosing the many body ground state |φ = |GS and the diagonalizing Majorana basis (p, q) for which the ground state is the vacuum state and the parity of j-mode is given by P j = −ip j q j , Equation (A2) can be expressed as follows: The covariance matrix in the (p, q) basis in Equation (A3) is very useful. Indeed, the first excited state can be obtained by Equation (A3) just exchanging a −1 with a 1 between the two off-diagonal blocks. In general, the covariance matrix for the thermal state is the following:Γ with ±λ = ± tanh(−β α /2). An analogous quantity to Equation (A1) can be defined for Bogoliubons which are related to the Majorana operators via the relation We are interested in evaluating the fermionic correlations which have an half as large a dimension as the fermionic covariance matrix in Equation (A6). So, defining the correlation matrix Λ: the desired relation holds: Λ ( f , f † ) = 1/2 (Γ ( f , f † ) + I) with I representing the identity matrix. The transformation of the covariance matrix picture in terms of Majornas (p, q) is then achieved by the action of the matrix (A8), where: i.e., where the following identity holds: It should be noted that Equation (A9) for a thermal state (Equation (A4)) returns the Fermi distribution: and so Below, we report the expressions of the previous matrix elements in terms of u and v coefficients whose expressions are known by means of Equation (A14) in terms of O and Ω matrices: where the repeated index k means a summation.