New scheme for braiding Majorana fermions

Non-Abelian statistics can be achieved by exchanging two vortices in topological superconductors with each grabbing a Majorana fermion (MF) as zero-energy quasi-particle at the cores. However, in experiments it is difficult to manipulate vortices. In the present work, we propose a way to braid MFs without moving vortices. The only operation required in the present scheme is to turn on and off local gate voltages, which liberates a MF from its original host vortex and transports it along the prepared track. We solve the time-dependent Bogoliubov–de Gennes equation numerically, and confirm that the MFs are protected provided the switching of gate voltages for exchanging MFs are adiabatic, which takes only several nano seconds given reasonable material parameters. By monitoring the time evolution of MF wave-functions, we show that non-Abelian statistics is achieved.

It was revealed that MFs appear inside vortex cores in topological p-wave SCs [7], and that non-Abelian statistics can be achieved by exchanging positions of vortices hosting MFs [28]. However, it is difficult to manipulate vortices in experiments, which may hinder the realization of this genius idea. To circumvent this problem, MFs at sample edges of topological SCs have been considered [31]. Making use of their topological properties, edge MFs can be braided with desired non-Abelian statistics by tuning point-like gate voltages on links among topological SC samples. In order to make the edge MFs stable, one needs to embed the device into a good insulator. The size of topological SCs should also be chosen carefully since the wave-functions of edge MFs Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. become too dilute for large samples, which makes edge MFs fragile due to excited states with small energy gap.
In this work we concentrate on MFs grabbed at vortex cores. We demonstrate that the core MFs can be liberated from vortex cores, transported and braided by applications of local gate voltages. The scheme takes fully advantages of SC/ SM/FMI heterostructure in the way shown schematically in figure 1: four holes are punched in the SM layer, and three electrodes are placed above the small regions between holes; gate voltages can be applied via the electrodes, and the ones at high voltage state (pink rectangular prisms in figure 1) connect holes by killing electron hoppings locally; one vortex (blue cylinders in figure 1) is induced and pinned right beneath each hole in the common SC substrate (SC and FMI are not shown explicitly for clarity). The key observation is that the geometric topology of the SM layer can be controlled by local gate voltages, and that when even number of holes are connected, core MFs fuse into quasi-particle states with finite energies, while one core MF exists when odd number of holes are connected. Core MFs can then be liberated from and transported among vortices with a sequence of turning on and off gate voltages at the electrodes. By solving the timedependent Bogoliubov-de Gennes (TDBdG) equation upon adiabatic tunings of gate voltages, we simulate the time evolution of MF wave-functions and confirm that the braiding of core MFs obeys non-Abelian statistics. Finally, we compare briefly our scheme of manipulating MFs with other proposals in both topological 2D + p p i SCs and 1D nanowire networks and show advantages of our scheme.

Tight-binding model
We start from a tight-binding Hamiltonian on square lattice for a SM with Rashba-type spin-orbit coupling in proximity to a FMI [

Hamiltonian in momentum space and Chern number
Before solving the BdG equation in a finite sample, we reveal the condition for achieving topological superconducting state in an infinite system. We transform Hamiltonian (1) into momentum space by expanding the annihilation operator as where N is the total number of lattice sites, k is the crystal momentum. We then obtain the momentum space Hamiltonian on the basis ↑  There are four holes in SM layer (yellow platform) with one superconducting vortex (blue cylinder) pinned right beneath each of them. The electrodes at high-voltage states (pink rectangular prisms) prohibit electron hoppings in the regions below them, and thus connect effectively the holes; the blue rectangular prism denotes an electrode at zero-voltage state.
we obtain the energy dispersion (see figure 2(a)) By squaring both sides of (7), we find that the conditions to close the bulk energy gap are Equation (8) can be fulfilled only when . Combining the two conditions in (8), we arrive at the critical V z to close the bulk energy gap at k-points where To investigate whether the system is topologically nontrivial when the bulk gap closes and reopens while tuning the Zeeman field V z , we evaluate the Chern number [18,19] is the wave-function of band n at momentum k. Integrating the Berry curvature  ⃗ × ⃗  shown in figure 2(b) over the Brillouin zone [33], we find that c = 1 for The nonzero Chern number is attributed to the topologically nontrivial energy gap at Γ point = k k ( , ) (0, 0) x y while those at π − (0, ), π − ( , 0) and π π − − ( , ) remain trivial [16,20,34]. The nonzero Chern number indicates that the system is topological and thus may support MFs. The above result is consistent with that obtained by the k p · approximation around Γ point [34].

Core MFs
Now we study a finite sample with two separated holes in the SM and two vortices of positive vorticity pinned right beneath them (see top panel of figure 3(a)). The typical size of a sample is 600 × 300 nm 2 , which is divided into 400 × 200 square grids, corresponding to the Hamiltonian matrix of dimension × 10 10 5 5 in (3). By solving the BdG equation for this case, we obtain the energy spectra of excitations and eigen-functions. Two zero-energy states are found at the holes, whereas no such state at the edge, as shown in figure 3(b). We examine the four spinor components of the zero-energy states, and find = ↑ ↑ u v * and = ↓ ↓ u v * (displayed explicitly in figure 3(c) for the right hole), which results in γ γ = † , indicating that the two zero-energy states are Majorana states.
It should be noticed that the excitation energy gap at vortices is about four times larger than that at the edge (see inset of figure 3(b)), which makes the core MFs more stable than edge MFs [31]. On the other hand, because the minigap associated with Andreev bound states at superconducting vortex core proximity-induced in SM is roughly with a small Fermi energy μ Δ ∼ [35], the influence from Andreev bound states to the core MFs can be neglected. It is in contrast to the case of SC exposed to vacuum where μ Δ ≫ and thus the minigap is small in order of Δ μ 2 .
Next, we impose a point-like gate voltage on the region between the two holes to prohibit direct hopping of electrons by lifting the on-site energy there as in the bottom panel of figure 3(a) (see also figure 1). This merges effectively the two isolated holes into a unified one. Solving the BdG equation for this case, there is no zero-energy quasi-particle, since the combined hole includes two vortices [36].

Transportation of MFs
Based on the above result that two MFs can fuse into finite energy excitations by connecting two holes, we can design a way of liberating and transporting a MF from one vortex to another. Initially, the top and middle holes are connected together while the leftmost one is isolated and hosts a MF (see figure 4(a) with t = 0). We then combine these three holes by applying gate voltages on the region between the left and middle ones, which causes the MF to spread itself over the unified hole including three vortices (see figure 4(a) with t = T). Finally, the MF is moved totally to the top by disconnecting the top hole from others (see figure 4(a) with = t T 2 ). It is noticed that the collapsing of MF wave-function on the top hole is a topological property, and is impossible for electrons and photons. The energy gap remains finite during the whole processes, which guarantees topological protections on the MF state.

Braiding of MFs
Being able to transport one MF from one hole to another, we extend the scheme to interchange positions of two core MFs in the system shown in figures 1 and 4. Following the above transportation procedures, we further move the green MF from the right hole to the left one during = ∼ t T T 2 4 in the same way as above. Finally, the red MF stored temporarily at the top hole is transported to the right one in the period  6 . After the sequence of switching processes, the system comes back to the original state with the red and green MFs exchanged.
In order to keep the topological protection, we need to manipulate the gate voltage in an adiabatic way. The reason is that the MF states have certain probabilities to be excited to higher energy states for non-adiabatic processes, which results in the collapsing of the whole braiding scheme. Given reasonable material parameters, the typical time for a single round of braiding is estimated to be within several nano seconds, which is sufficiently short time for practical applications.

TDBdG equation
In order to investigate the impact of position exchanging to MF states, we monitor the time evolution of MF wavefunction Ψ 〉 t | ( ) by solving the TDBdG equation numerically where H(t) is given in (3) and depends on time in terms of the hopping rates t ij and α t i at the regions between holes, which are tuned adiabatically by the local gate voltages [31].
Even with the powerful computation resources available these days, it is still hopeless to tackle this problem by directly diagonalizing the Hamiltonian H(t) of dimension × 10 10 6 6 for each time instant. Fortunately, it has been revealed that when the exponential operator is expanded by the Chebyshev polynomial , which reduces the computation cost further. In this way, the TD wave-function of the MFs can be obtained efficiently with sufficient accuracy in an iterative fashion Ψ δ δ Ψ R the zero-energy states at left and right holes, respectively, as shown in figures 4(a) and (b). We evaluate the projections of during adiabatical braiding processes and display them in  . This indicates that the braiding of MFs satisfies non-Abelian statistics [24,25,31,40,41,43,44], as will be shown explicitly below.

Braiding MFs in 2D p-wave SCs
We begin by reviewing Ivanovʼs model for realizing non-Abelian statistics, which describes low-energy excitations bound to vortices in spinless p-wave SCs [28]. The Hamiltonian is [28][29][30] where the first term describes kinetic energy and chemical potential ε − F , c † is electron creation operator, and the second term gives pairing function Δ r ( ) with superconducting phase θ, k x and k y are electron momenta along x-and y-axis respectively. The Bogoliubov quasi-particle operator is , which satisfies the BdG equation † † with E the eigenenergy of quasi-particle. By taking Hermitian conjugation † at both sides of (15), we have indicating the particle-hole symmetry of BdG equation, i.e.
. Thus, the zero energy states must be doubledegenerate, satisfying self-adjoint condition γ , which are MFs. We first consider how MFs evolve under U (1) gauge transformation. Since θ can be absorbed into fermionic creation and annihilation operators: = θ c c e † i 2 † and Figure 5. Projections of the MF wave-function Ψ 〉 t | ( ) obtained by TDBdG onto the initial states c c e i 2 , the new Bogoliubov quasi-particle operator is In the case of π 2 phase changing θ θ π → + 2 , we have γ γ = − † † , which is an important result that can be employed to achieve non-Abelian statistics.
In a 2D spinless + p p i SC with many vortices, superconducting phase θ around each vortex can be single-valued apart from a cut (dashed black line in figure 6), where θ jumps by π 2 . It is then easy to see that the Majorana wave-function bound to a vortex core picks up a sign γ γ → − after crossing the cut. By interchanging positions of vortices i and + i 1 (see red arrows in figure 6), one observes that MF γ i passes through the cut of vortex + i 1 and picks up π 2 phase (see figure 6), which gives  with T i the braiding operation, consistent with our results of braiding MFs (13). (20) is given by [28] [3]. Direct manipulations of vortices might be done by using a STM tip, but suffer great difficulties. This is not only because that large effective masses of vortices make transportations of themselves hard and time-consuming, but also it is almost impossible to return vortices to exact positions in order to form a closed braiding loop, which would cause systematic errors in topological quantum computations. On the contrary, in our scheme of braiding MFs, vortices stay pinned at their original positions and thus no motion of vortices is necessary. What we do is tuning gate voltages at small regions connecting holes during braiding processes (see figures 3(a) and 4). It is guaranteed that MFs exchange their positions exactly after braiding.

Braiding MFs in a nanowire network
A toy model for realizing MFs in a N-site 1D spinless p-wave SC was proposed by Kitaev [8] h. c. , (24) with μ the chemical potential, t nearest neighbor hopping rate, Δ superconducting gap function with phase θ. In the special Figure 6. Braiding of vortex i and vortex + i 1 in a 2D spinless pwave superconductor. Dashed black lines connecting to bottom boundary are cuts for superconducting vortices [28]. Red arrows are braiding loops. with finite energy. The number operator of fermion = n f f † has double-degenerated groundstates 〉 |0 and 〉 |1 for ϵ → 0. An energy gap ϵ is created between eigenstates 〉 |0 and 〉 |1 for finite ϵ. In order to exchange MFs, three nanowires are put close to each other, and MFs at positions C1 and C2 are allowed to couple together, forming a complex fermion with finite energy (see figure 8(a)). To transport MF γ 2 , the authors adiabatically reduce the coupling between MFs at C1 and C2, and then increase that between B1 and C1. At zero coupling between C1 and C2, γ 2 is moved to position C2 (see figure 8(b)). Next, γ 1 at A2 is transported to B1 (see figure 8(c)). Finally, γ 2 is moved to position A2 to complete the interchanging of MFs (see figure 8(d)). The trajectories followed by γ 1 and γ 2 are It is proven that interchanging of γ 1 and γ 2 in this way follows non-Abelian statistics since either γ 1 or γ 2 acquires a minus sign after exchanging positions [24]. In order to realize the above two braiding methods in nanowire networks [24,41], some difficulties need to be overcome. In the proposal by Alicea et al [41], manipulations of MFs by using gate voltages become quite difficult in 1D system since one has to adjust a number of local gate voltages precisely in the whole network. Failing to exert correct gate voltage at a single lattice site may break down the whole braiding process. Besides, transportations of MFs through Tjunctions depend on details of the junction, which may be difficult to control [24,38]. As for the scheme by Sau et al, the coupling ϵ between MFs is oscillatory in space in topological SCs as well as upon changing chemical potentials [39], which is not easy to be controlled accurately in any macroscopic way, meaning that transportations of MFs can hardly be carried out in designed ways. In contrast, our proposal for manipulating MFs only requires tuning of local gate voltages at very small regions connecting two holes (see figure 3) and does not involve microscopic control on the couplings between MFs, which makes our scheme of braiding MFs robust.

Conclusion
We have shown that the MFs hosted by vortex cores in topological SCs can be liberated from pinned vortices, transported and braided over the prepared holes, taking advantages of the heterostructure of s-wave SC and spin-orbit coupled SM. By solving the TDBdG equation numerically, we monitor the time evolutions of MF wave-functions and demonstrate the non-Abelian statistics of adiabatical braidings of MFs. The present scheme only requires local applications of gate voltages, and minimizes possible disturbances to the MFs, which might be a challenging issue in other proposals based on end MFs in 1D SCs where gate voltages are necessary along the whole system. Instead of exchanging positions of vortices hosting MFs, our scheme of braiding operation is much easier and faster for experimental realizations of non-Abelian statistics. As compared with the edge MFs in 2D topological SCs, the core MFs are protected by a larger energy gap, which relaxes the limitation on operating temperatures. Therefore, the present scheme provides a more feasible way for manipulating MFs.