Majorana zero modes in the hopping-modulated one-dimensional p-wave superconducting model

We investigate the one-dimensional p-wave superconducting model with periodically modulated hopping and show that under time-reversal symmetry, the number of the Majorana zero modes (MZMs) strongly depends on the modulation period. If the modulation period is odd, there can be at most one MZM. However if the period is even, the number of the MZMs can be zero, one and two. In addition, the MZMs will disappear as the chemical potential varies. We derive the condition for the existence of the MZMs and show that the topological properties in this model are dramatically different from the one with periodically modulated potential.

one MZM. However if the period is even, in some parameter regimes the number of the MZMs can be two. Furthermore, the MZMs will disappear as the chemical potential varies no matter the period is odd or even. Therefore the topological properties of the hopping-modulated model are drastically different from those of the potential-modulated one.

Method
We consider a one-dimensional Kitaev p-wave SC model where the hopping is periodically modulated, the Hamiltonian can be written as  . Since we assume t i and Δ in Eq. (1) is real (up to a global phase) throughout the paper, thus H k respects the time-reversal, particle-hole and chiral symmetries and it can be unitarily transformed to an off-diagonal matrix as [26][27][28] here τ y is a Pauli matrix acting on the particle-hole space. Then the system belongs to the class BDI which is characterized by the  index while the number of the MZMs can be represented by W which is calculated through In fact, W just counts how many times the determinant of A k crosses the imaginary axis as k evolves from 0 to π/q. Since Det(A k ) is real, W must be zero otherwise Det(A k ) will be zero for some k (which means the bulk energy gap vanishes), therefore, there can be no MZMs.

Results and Discussion
Generally, the periodic modulation can take many forms. For arbitrary t 1 and t 2 (t 1 ≠ − t 2 ), we have , then Det(A k ) will cross the imaginary axis once as k changes from 0 to π/2. In this case, = W 1 and one MZM exists. Interestingly, at V = 0, we have In this case, the system can be divided into two separated subsystems A 1k and A 2k . As k evolves from 0 to π/2, we have Scientific RepoRts | 5:17049 | DOI: 10.1038/srep17049 Therefore, if both 2 are less than zero, then two MZMs will show up in this case and the existence of these two MZMs has been numerically verified (for example, t 1 = 0.5, t 2 = − 0.8 and Δ = 0.5). For where t 1 = cos(2π/3 + δ), t 2 = cos(4π/3 + δ) and t 3 = cosδ. In this case, (− and + are for k = 0 and π/3, respectively). If , then ( ) Det A k will cross the imaginary axis exactly once as k evolves from 0 to π/3, which means that = W 1 and one MZM exists in this case [E 1 in Eq. (2) is zero under OBC]. On the contrary, means that the bulk energy gap vanishes and ( ) ( means that Det(A k ) will not cross the imaginary axis as k evolves from 0 to π/3 in such a way that W = 0, in both cases no MZMs exist. Specifically, for δ = (2m + 1)π/6 with m being an integer, we have cos3δ = 0, therefore ( ) and no MZMs can exist, irrespective of the values of Δ and V. For example, we set Δ = 1 and L = 1632. In Figs 1 and 2 we plot the energy spectra under OBC and ( ) 3 under PBC, respectively. As we can see, at V = 0, MZM exists for any δ, except for δ = π/6, π/2, 5π/6, 7π/6, 3π/2, 11π/6 where the bulk energy gap closes. As V increases, for some δ, MZM vanishes and as δ evolves from 0 to 2π, topologically trivial (without MZM) and nontrivial (with one MZM) phases appear in turn. A typical distribution of the zero-mode MFs is shown in Fig. 3 and we can see that the two MFs γ i A and γ i B are well separated in real space and are located at the left and right ends, respectively while the actual decay length of these two MFs increases/decreases as the bulk energy gap decreases/increases.  At first glance, since Det(A k ) is real, thus, similar to the α = 1/2 case, there should be no MZMs. However, this is not true at V = 0. In the following we set Δ = 1 and L = 1632 as an example. As we can see from Fig Fig. 1(d). and the distribution of the zero-mode MFs is shown in Fig. 5. The existence of these two MZMs can be explained as follows. At V = 0, we found that the eigenvalues of H k in Eq.   At . If δ ( ∆ − ) < cos 2 2 1 0 2 , then both Det(A 1k ) and Det(A 2k ) will cross the imaginary axis exactly once as k evolves from 0 to π/4, indicating that there exists one MZM in each subsystem and the number of the MZMs for the whole system is two. Therefore for α = 1/4, at V = 0, the number of the MZMs are either two or zero while at V ≠ 0, there are no MZMs. For general t i ( = , … , ) i 1 4 , at V ≠ 0, Det(A k ) may not be real and there may exist one MZM. However at V = 0, the system can still be divided into two subsystems. In this case, if the conditions are satisfied simultaneously, there will be two MZMs.
Furthermore we found that, for general periodic modulation, if the period q is odd, then the number of the MZMs is either zero or one. On the other hand, if q is even, then at V ≠ 0, the number of the MZMs is still zero or one. However at V = 0, the system can always be divided into two independent subsystems and if the conditions are simultaneously satisfied, there will be two MZMs.
In summary, we have studied the number of the MZMs and their stability in the hopping-modulated one-dimensional p-wave SC model. We found that the former strongly depends on the period of the modulation. If the period q is odd, there can be at most one MZM in the system while for an even q, the number of the MZMs can be zero, one and two. The existence of two MZMs can occur only at V = 0, since in this case, A k in Eq. then after a tedious calculation we can prove that Det(C 1k ) and Det(C 2k ) cannot be complex simultaneously and they can cross the imaginary axis at most once as k varies from 0 to π/q. Therefore, even if A 1k Scientific RepoRts | 5:17049 | DOI: 10.1038/srep17049 can be further separated into two subblocks C 1k and C 2k , only one of them may host one MZM, making the maximal number of the MZMs in A 1k be one. The same argument can be applied to A 2k as well. Thus for an even modulation period, there can be at most two MZMs. For the specific modulation form we considered [t i = cos(2πiα + δ) with α = p/q], only at q = 4n ( = , , , …) n 1 2 3 can Eqs (19) and (20) be simultaneously satisfied, therefore only in this case can there exist two MZMs. Furthermore, the MZMs will vanish as the chemical potential V varies. In the periodically potential-modulated model considered in Refs. 21-23, when the time-reversal symmetry is present, there can be at most one MZM and if the potential vanishes at certain sites, then the MZM will be very robust and stable for arbitrary strength of the modulation. Clearly this is not the case in the periodically hopping-modulated model, therefore the topological properties differ drastically between these two models.
At last we would like to emphasize the motivation as well as the physical implications of our study. As we know, exploring various topological properties in different models is of both fundamental and practical importance. From the fundamental point of view, it may help people to understand the mechanism and condition for the existence of the MZMs. As stated in the introduction section, intuitively people may speculate that the topological properties are similar between the hopping-modulated and potential-modulated Kitaev models. However in fact this is not the case as has been demonstrated in our study where both the number and stability of the MZMs differ drastically between these two models and these different behaviors have never been reported before. Furthermore we have demonstrated that, for multiband systems, special caution has to be taken when calculating the number of the MZMs from the  index. That is, when the system can be separated into two subsystems, the number of the MZMs may be mistakenly thought to be zero while there are actually two MZMs. On the other hand, from the practical point of view, our work, together with those previous studies concentrating on the potential modulation, may help to guide researchers to fabricate various topological phases with different numbers of the MZMs and to further manipulate them in order to realize topological quantum computation. We expect that our model is most likely to be realized in cold-atom systems and in optical superlattices where the hopping can be adjusted. In solid state devices, the direct modulation of hopping may be difficult. However we notice, the p-wave Kitaev  and h = − V/2. Therefore the chemical potential V = 0 can be achieved by setting h = 0 (zero Zeeman field). Furthermore, since t = J x + J y and Δ = J x − J y , the modulation of hopping may be possible if J x + J y varies in space while J x − J y is constant. This may be realized in atomic chains with a spatially modulated spin arrangement (see refs. [16][17][18][19][20]. Therefore the ideas in our work are both fundamentally sound and practically applicable.