Majorana fermions are particles which are their own antiparticles and are expected to obey non-Abelian statistics.^[1,2] Recently, there is a growing interest in searching for Majorana fermions in the condensed matter physics community.^[3] Majorana zero modes (MZMs) are quasiparticle excitations in condensed matter systems and are expected as building blocks of realizing the fault-tolerant topological quantum computation protected from local decoherence.^[4–8] Many theoretical schemes have been proposed to create MZMs in the condensed matter physics systems.^[9–19] Several experiments of detecting the signatures of MZMs in semiconductor nanowires with proximity-induced superconductivity^[20–23] and iron atomic chains on the surface of the superconducting lead^[24] have been reported.

In Kiteav’s pioneering work,^[25] it was found that two unpaired MZMs can be localized at the ends of a one-dimensional (1D) spinless p-wave superconductor (i.e., the so-called Kiteav chain). The Kitaev chain is the simplest model that could realize the MZMs, thus the topological properties of the Kiteav chain have attracted considerable interest.^[26–36] The Kitaev chain has been investigated under the effect of p-wave pairing,^[26] parametric pumping,^[28] periodically modulated chemical potential,^[31] periodically modulated hopping,^[32] and classical noise,^[33] respectively. The topological order^[34] and decoherence^[30] in interacting Kitaev chains were also considered. It is proposed that the Kitaev chain system may exhibit the Zitterbewegung oscillations.^[35] Wakatsuki et al. studied theoretically the topological phase transitions and fermion fractionalization to Majorana fermions in a 1D dimerized Kitaev superconductor model, which is a hybrid system comprised of the Su–Schrieffer–Heeger (SSH) model^[37] and the Kitaev model.^[25] Very recently, Liu investigated the effects of the next-nearest-neighbor hopping and nearest-neighbor interactions on topological phases in a 1D dimerized Kitaev chain.^[36]

Note that Potter and Lee generalized 1D Kitaev chain to a quasi-1D ladder model of the Kitaev p_x + ip_y superconductor chains.^[38] They indicated that the MZMs can be localized at opposite ends of a spinless p-wave superconductor with the strip geometry. A similar set of ideas on Majorana fermions in the quasi-1D wires was discussed by Wimmer et al.^[39] Later, Zhou and Shen re-examined an ideal spinless quasi-1D ladder model of the Kitaev p_x + ip_y superconductor chains, studied the topological quantum phase transition in this system and gave the phase diagrams of the presence of MZMs in quasi-1D sample by use of ℤ₂ topological index.^[40] Up to now, the studies on MZMs in the quasi-1D ladder model of the Kitaev superconductor chains, multichannel spinless p-wave superconducting wires and multiband semiconducting nanowires have also attracted much attention.^[41–47]

In this paper, we focus on a ladder model of the dimerized Kitaev superconductor chains. We discuss two topological classes of this model and calculate their corresponding topological number ℤ and ℤ₂. The phase diagrams with respect to model parameters are demonstrated, and the topological phase transitions in the two distinct topological classes are illustrated.

This paper is organized as follows. In Section 2, we introduce a ladder model of the dimerized Kitaev chains and discuss the topological class of the model. In Section 3, in order to discuss topological phase transition of the class BDI, we calculate the ℤ index winding number W and obtain the phase diagrams. In Section 4, we focus our attention on the case of the class D, and the ℤ₂ index Majorana number ℳ is introduced to discuss topological phase transition of the class D. Finally, we summarize our results in Section 5.

We consider a ladder model of the dimerized Kitaev chains of spinless p-wave superconductors, the illustration of this model is shown in Fig. 1, which is described by the following tight-binding Hamiltonian

Fig. 1.

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	Fig. 1. Illustration of the model. The blue and yellow circles present two sublattices (A and B). The (thick and thin) solid lines indicate the dimerized Kitaev chain, and the dotted lines indicate the inter-chain coupling. α denotes the α-th Kitaev chain along the y axis, and j denotes the j-th unit cell (presented by green oval) containing two sites along the x axis.

Here we assume that the number of the dimerized Kitaev chains is n in the ladder model, and the number of unit cells in each chain is L, that is, the length of lattice site is 2L along the x-axis direction. We note that the system is reduced to a one-dimensional dimerized Kitaev superconductor model when n = 1.^[27] Now, one introduces a periodic boundary condition along the x-axis direction, i.e., , and uses the Fourier transform of the operator ,

The topological class of the Hamiltonian (1) can be clarified according to its behavior under time-reversal symmetry, particle–hole symmetry, as well as chiral symmetry.^[48,49] The time-reversal (Θ), the particle–hole (Ξ), and the chiral (Π) operators are defined by Θ = K, Ξ = τ_xK, and Π = τ_x, respectively,^[43] where K represents the complex conjugate operator and τ_x is the Pauli matrix in the particle–hole space. We note that the Hamiltonian (1) reduces to a ladder model of the conventional Kitaev chains as the dimerization parameter η = 0.^[43] Wakatsuki et al. have indicated that the ladder model without dimerization belongs to the class BDI when the phase difference θ is 0 or π. Otherwise, it becomes the class D. For the ladder model of the dimerized Kitaev chains described by the Hamiltonian (1), we can check that when θ = 0 or π the Hamiltonian (1) still satisfies the following relations: ΘH(k)Θ⁻¹ = H(−k), ΞH(k)Ξ⁻¹ = −H(−k), and ΠH(k)Π⁻¹ = −H(k). Thus, the Hamiltonian (1) with θ = 0 or π still belongs to the class BDI. When the phase difference is θ ≠ 0 or π, we can also check that both the time-reversal symmetry and the chiral symmetry are broken while the particle–hole symmetry is present, so, in this case the topological class of the Hamiltonian (1) is the class D.

In what follows, we will investigate the topological phase transition in the two different topological classes, which is induced by the variation of the superconducting pairing phase difference θ between |Δ_x| and |Δ_y|.

We first discuss the case of θ = 0. As mentioned above, the system belongs to the class BDI which is characterized by the ℤ index. In order to analyze the topological property of the system, we introduce a unitary transformation

Fig. 2.

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Fig. 2. Phase diagram for the ladder model of the dimerized Kitaev chains as a function of the pairing amplitude and the dimerization with chain numbers n along the y axis. The energy unit is tx, and we have taken θ = 0, |Δx| = |Δy| = Δ, ty = 0.5tx, and μ = 0.5tx. The integer numbers in the figure represent the ℤ index. The white dashed lines guide the value of Δ = 0.1tx. (a) n = 2; (b) n = 3; (c) n = 4; (d) n = 5; (e) n = 6; (f) n = 7.

The topological phase diagrams with t_y/t_x = 0.3 and η/t_x = 0.3 in the (Δ,μ) plane are shown in Fig. 3. It is found that when Δ/t_x < 0.3 (cf. Δ/t_x = 0.1) along the μ axis multiple topological phase transitions occur, and the ℤ index increases from 0 to n by 1 at every phase transition with the increasing μ. On the other hand, along the Δ axis with μ = 0, the topological phase transition occurs from the ℤ index W = 0 to W = n at Δ/t_x = 0.3.

Fig. 3.

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Fig. 3. Phase diagram for the ladder model of the dimerized Kitaev chains as a function of the pairing amplitude and the chemical potential with chains numbers n along the y axis. The energy unit is tx, and we have taken θ = 0, |Δx| = |Δy| = Δ, ty = 0.3tx, and η = 0.3tx. The integer numbers in the figure represent the ℤ index. The white dashed lines guide the value of Δ = 0.1tx. (a) n = 2; (b) n = 3; (c) n = 4; (d) n = 5; (e) n = 6; (f) n = 7.

According to the topological argument, the topologically nontrivial phase corresponds to the existence of the zero energy mode, and the ℤ index gives the number of MZMs, which can be justified by the excitation spectrum obtained by the numerical diagonalization of the Hamiltonian Eq. (1). The energy spectra of the ladder model of the dimerized Kitaev chains with respect to η and μ are plotted in Figs. 4 and 5, respectively. For sufficiently long but finite chains, one can find spatially isolated zero-energy Majorana end states localized at opposite ends of the chains, thus each zero-energy state per chain doubly degenerates due to the two endpoints of the chain. In Fig. 4, numerical diagonalizations show that as η increases, the number of zero-energy decreases by 2 at every phase transition; the energy spectra shown in Fig. 5 present that as μ increases, the number of zero-energy increases by 2 at every phase transition. The results given in Figs. 4 and 5 confirm the topological properties characterized by the ℤ index shown in Figs. 2 and 3 (cf. white dashed line guiding the value of Δ/t_x = 0.1).

Fig. 4.

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	Fig. 4. Energy spectra with respect to η for the finite chains obtained by numerical diagonalization of the Hamiltonian (1) with parameters: L = 128, θ = 0, ty/tx = 0.5, μ/tx = 0.5, and Δ/tx = 0.1. The numbers of chains are (a) n = 3; (b) n = 4; (c) n = 5; (d) n = 6.

Fig. 5.

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	Fig. 5. Energy spectra with respect to μ for the finite chains obtained by numerical diagonalization of the Hamiltonian (1) with parameters: L = 128, θ = 0, ty/tx = 0.3, η/tx = 0.3, and Δ/tx = 0.1. The numbers of chains are (a) n = 3; (b) n = 4; (c) n = 5; (d) n = 6.

Now we focus our attention on the case with θ = π/2, in which the topological class of Hamiltonian (1) belongs to the class D and is characterized by the ℤ₂ index. In the Majorana representation, we define a set of the operators

According to Eq. (16), we numerically calculate Majorana number ℳ as a function of physical parameters and then plot the phase diagrams showing a sequence of topological phase transition for the different numbers n of the dimerized Kitaev chains. The topological phase diagrams of the class D with θ = π/2 in the (η,μ) plane for the even and odd numbers of chain are shown in Fig. 6, where we have taken |Δ_x| = |Δ_y| = 0.8t_x and t_y = t_x. We now analyze these phase diagrams. First, it is observed that for the given value of dimerization η, the topologically nontrivial (ℳ = −1) and trivial phases (ℳ = +1) alternate with the variation of the value of μ. Take the case with the number of chains n = 5 (cf., Fig. 6(d)) as an example, along the μ axis with η/t_x = 0.2, ℳ = −1 for | μ |/t_x < 1, ℳ = +1 for 1 < | μ |/t_x < 1.42, and again ℳ = −1 for 1.42 < | μ|/t_x < 1.96, then again ℳ = +1 for 1.96 < |μ |/t_x < 2.62, and again ℳ = −1 for 2.62 < | μ |/t_x < 2.96, up to ℳ = +1 for | μ |/t_x > 2.96. Additionally, it is shown that topologically nontrivial regions in the (η,μ) plane generally shrink with increasing dimerization η. Namely, the topological phase of the class D characterized by the ℤ₂ index is also suppressed by the dimerization.

We note also an interesting property from the phase diagrams. For some given parameters, the topological property of the ladder of the dimerized Kitaev chains will show even–odd effects with respect to the number n of chains. For instance, in Fig. 6, when the parameters η/t_x = 0.2 and μ/t_x = 0.5 are fixed, the topologically nontrivial and trivial phases alternate when the chain numbers n change, ℳ = +1 for even n (n = 2,4,6) and ℳ = −1 for odd n (n = 3,5,7).

We show energy spectra with finite chains as functions of μ in Fig. 7, where we have set t_y/t_x = 1.0, η/t_x = 0.2, and |Δ_x|/t_x = |Δ_y|/t_x = 0.8. Our numerical results confirm that there are MZMs in topologically nontrivial regions presented in phase diagrams of Fig. 6 (cf., red dashed lines along the μ axis with η/t_x = 0.2), while MZMs disappear in topologically trivial regions. Note that the class D is characterized by the ℤ₂ index, thus for sufficiently long but finite chains, one can find a pair of MZMs in topologically nontrivial phases due to the two endpoints of every chain. Further, the even–odd effects are not only observed in phase diagrams of Fig. 6, but also appear in the energy spectra of Fig. 7.

Fig. 6.

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Fig. 6. Phase diagram for the ladder model of the dimerized Kitaev chains as a function of the dimerization and the chemical potential with chain numbers n along the y axis. The energy unit is tx, and we have taken θ = π/2, |Δx| = |Δy| = 0.8tx, and ty = tx. Blue region denotes the topologically nontrivial phase (ℳ = −1), and yellow region denotes the topologically trivial phase (ℳ = +1). The red dashed lines guide the value of η = 0.2tx. (a) n = 2; (b) n = 3; (c) n = 4; (d) n = 5; (e) n = 6; (f) n = 7.

Fig. 7.

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	Fig. 7. Energy spectra with respect to μ for the finite chains obtained by numerical diagonalization of the Hamiltonian (15) with parameters: L = 512, θ = π/2, ty/tx = 1.0, η/tx = 0.2, and |Δx|/tx = |Δy|/tx = 0.8. The numbers of chains are (a) n = 2; (b) n = 3; (c) n = 4; (d) n = 5.

In this paper, we investigate the topological phase transition of a ladder model of the dimerized Kitaev superconductor chains. The phase difference θ between the inter- and intra-chain superconducting pairings causes the change of topological class of the system between the class BDI and the class D. The system with θ = 0 or π belongs to the class BDI characterized by the ℤ index, and the other is the class D characterized by the ℤ₂ index. When the phase difference θ ≠ 0 or π, the topological class of the system is the class D characterized by the ℤ₂ index. By calculating two different topological numbers, i.e., the ℤ index winding number W and the ℤ₂ index Majorana number ℳ, we illustrate the topological phase diagrams of the system. It is observed that the system with θ = 0 belonging to the class BDI undergoes multiple topological phase transitions accompanying the variation of the number of MZMs. In the case of θ = π/2, the system belongs to the class D, it is shown that for the given value of dimerization, the topologically nontrivial and trivial phases alternate with the variation of the value of chemical potential. Furthermore, the even–odd effect is also observed in phase diagrams of the class D. Additionally, it is indicated that the topological phase with MZMs is suppressed by the dimerization. The topological properties presented in phase diagrams are consistent with the results of the energy spectra obtained by numerical diagonalization.