New scheme for braiding Majorana fermions

We start from a tight-binding Hamiltonian on square lattice for a SM with Rashba-type spin–orbit coupling in proximity to a FMI [31]

$$\begin{eqnarray}\begin{array}{rcl} {{H}_{0}} & = & -\mathop{\sum }\limits_{{\bf i},{\bf j},\sigma }{{t}_{{\bf ij}}}\hat{c}_{{\bf i}\sigma }^{\dagger }{{{\hat{c}}}_{{\bf j}\sigma }}-\mu \mathop{\sum }\limits_{{\bf i},\sigma }\hat{c}_{{\bf i}\sigma }^{\dagger }{{{\hat{c}}}_{{\bf i}\sigma }}+\mathop{\sum }\limits_{{\bf i}}{{V}_{z}}\left( \hat{c}_{{\bf i}\uparrow }^{\dagger }{{{\hat{c}}}_{{\bf i}\uparrow }}-\hat{c}_{{\bf i}\downarrow }^{\dagger }{{{\hat{c}}}_{{\bf i}\downarrow }} \right) \\ {} & {} & +\mathop{\sum }\limits_{{\bf i},\sigma ,\sigma ^{\prime} }\left\{ {\rm i}{{t}_{\alpha {\bf i}}}\left[ \hat{c}_{({\bf i}+\hat{x})\sigma }^{\dagger }\hat{s}_{y}^{\sigma \sigma ^{\prime} }{{{\hat{c}}}_{{\bf i}\sigma ^{\prime} }} \right. \right. \\ {} & {} & -\left. \left. \hat{c}_{({\bf i}+\hat{y})\sigma }^{\dagger }\hat{s}_{x}^{\sigma \sigma ^{\prime} }{{{\hat{c}}}_{{\bf i}\sigma ^{\prime} }} \right]+{\rm h}.{\rm C}. \right\}, \\ \end{array}\end{eqnarray} \tag{ 1 }$$

where ${{t}_{{\bf ij}}}$ and ${{t}_{\alpha {\bf i}}}$ are the nearest-neighbor hopping rates of electrons with preserved and flipped spin directions respectively; μ and V_z are chemical potential and strength of Zeeman field respectively; $c_{{\bf i}\sigma }^{\dagger }$ creates one electron with spin σ = ↑, ↓ at lattice site ${\bf i}$; $\vec{s}=({{\hat{s}}_{x}},{{\hat{s}}_{y}},{{\hat{s}}_{z}})$ are the Pauli matrices for spin; $\hat{x}$ and are unit lattice vectors along x and y directions.

The proximity-induced superconductivity in SM is described by

$$\begin{eqnarray}&&{{H}_{{\rm sc}}}=\mathop{\sum }\limits_{{\bf i}}\left( {{\Delta }_{{\bf i}}}\hat{c}_{{\bf i}\uparrow }^{\dagger }\hat{c}_{{\bf i}\downarrow }^{\dagger }+{\rm h}.{\rm C}. \right),\end{eqnarray} \tag{ 2 }$$

where ${{\Delta }_{{\bf i}}}$ is the pairing potential at site ${\bf i}$. The Bogoliubov–de Gennes (BdG) equation of total Hamiltonian $H={{H}_{0}}+{{H}_{{\rm sc}}}$ is given by

$$\begin{eqnarray}\left( \begin{array}{ccccccccccccccc} {{H}_{0}} & \Delta \\ {{\Delta }^{\dagger }} & -{{{\hat{\sigma }}}_{y}}H_{0}^{*}{{{\hat{\sigma }}}_{y}} \\ \end{array} \right)\Psi (\vec{r})=E\Psi (\vec{r}),\end{eqnarray} \tag{ 3 }$$

where $\Psi (\vec{r})\;={{\left[ {{u}_{\uparrow }}(\vec{r}),{{u}_{\downarrow }}(\vec{r}),{{v}_{\downarrow }}(\vec{r}),-{{v}_{\uparrow }}(\vec{r}) \right]}^{T}}$ is the Nambu spinor, and ${{\gamma }^{\dagger }}=\int {\rm d}\vec{r}{{\sum }_{\sigma }}{{u}_{\sigma }}(\vec{r})\hat{c}_{\sigma }^{\dagger }(\vec{r})+{{v}_{\sigma }}(\vec{r}){{\hat{c}}_{\sigma }}(\vec{r})$ is the quasi-particle operator. For a typical square sample with 200 × 200 sites, the Hamiltonian matrix in (3) has a dimension of ${{10}^{5}}\times {{10}^{5}}$. In most cases only several states near E = 0 are relevant as far as MFs are concerned, for which a powerful numerical technique is available [32].

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

$$\begin{eqnarray}&&{{\hat{c}}_{{\bf i}\sigma }}=\frac{1}{\sqrt{N}}\mathop{\sum }\limits_{{\bf k}}{{c}_{{\bf k}\sigma }}{{{\rm e}}^{{\rm i}{\bf k}\cdot {\bf i}}},\end{eqnarray} \tag{ 4 }$$

where N is the total number of lattice sites, ${\bf k}$ is the crystal momentum. We then obtain the momentum space Hamiltonian on the basis ${{[{{c}_{{\bf k}\uparrow }},{{c}_{{\bf k}\downarrow }},{{c}^{\dagger }}{{_{-{\bf k}\uparrow }}^{{}}},{{c}^{\dagger }}{{_{-{\bf k}\downarrow }}^{{}}}]}^{T}}$

$$\begin{eqnarray}&&\begin{array}{l} H({\bf k}) \\ =\quad \left( \begin{array}{ccccccccccccccc} \varepsilon ({\bf k})+{{V}_{z}}{{{\hat{\sigma }}}_{z}}+\vec{R}({\bf k})\cdot \vec{s} & {\rm i}\Delta ({\bf k}){{{\hat{\sigma }}}_{y}} \\ -{\rm i}{{\Delta }^{*}}({\bf k}){{{\hat{\sigma }}}_{y}} & -\varepsilon ({\bf k})-{{V}_{z}}{{{\hat{\sigma }}}_{z}}+\vec{R}({\bf k})\cdot {{{\vec{s}}}^{*}} \\ \end{array} \right), \\ \; \\ \end{array}\end{eqnarray} \tag{ 5 }$$

where $\varepsilon ({\bf k})=-2{{t}_{0}}({\rm cos} {{k}_{x}}+{\rm cos} {{k}_{y}})-\mu$ and $\vec{R}({\bf k})=(-2{{t}_{\alpha }}{\rm sin} {{k}_{y}},2{{t}_{\alpha }}{\rm sin} {{k}_{x}},0)$. Diagonalizing Hamiltonian (5), we obtain the energy dispersion (see figure 2(a))

$$\begin{eqnarray}&&\begin{array}{l} E({\bf k}) \\ \quad =\pm \sqrt{{{\varepsilon }^{2}}+|\vec{R}{{|}^{2}}+V_{z}^{2}+|\Delta {{|}^{2}}\pm 2\sqrt{V_{z}^{2}|\Delta {{|}^{2}}+{{\varepsilon }^{2}}(V_{z}^{2}+|\vec{R}{{|}^{2}})}}, \\ \; \\ \end{array}\end{eqnarray} \tag{ 6 }$$

with gap closed at

$$\begin{eqnarray}&&{{\varepsilon }^{2}}+|\vec{R}{{|}^{2}}+V_{z}^{2}+|\Delta {{|}^{2}}=2\sqrt{V_{z}^{2}|\Delta {{|}^{2}}+{{\varepsilon }^{2}}\left( V_{z}^{2}+|\vec{R}{{|}^{2}} \right)}.\end{eqnarray} \tag{ 7 }$$

By squaring both sides of (7), we find that the conditions to close the bulk energy gap are

$$\begin{eqnarray}&&\left\{ \begin{array}{ccccccccccccccc} |\Delta |\cdot |\vec{R}|=0, \\ {{\varepsilon }^{2}}+|\Delta {{|}^{2}}=V_{z}^{2}+|\vec{R}{{|}^{2}}. \\ \end{array} \right.\end{eqnarray} \tag{ 8 }$$

Equation (8) can be fulfilled only when $|R|=0$ (since $|\Delta |\gt 0$). The ${\bf k}$-points with $|R|=0$ are $({{k}_{x}},{{k}_{y}})=(0,0),(0,-\pi ),(-\pi ,0)$ or $(-\pi ,-\pi )$ in the first Brillouin zone ${{T}^{2}}=[-\pi ,\pi )\otimes [-\pi ,\pi )$. Combining the two conditions in (8), we arrive at the critical V_z to close the bulk energy gap at ${\bf k}$-points where $|R({\bf k})|=0$

$$\begin{eqnarray}&&V_{z}^{2}={{\left[ 2{{t}_{0}}({\rm cos} {{k}_{x}}+{\rm cos} {{k}_{y}})+\mu \right]}^{2}}+|\Delta {{|}^{2}}.\end{eqnarray} \tag{ 9 }$$

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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]

$$\begin{eqnarray}&&c=\frac{1}{2\pi {\rm i}}{{\int }_{{{T}^{2}}}}{{{\rm d}}^{2}}{\bf k}\cdot \left( \vec{\nabla }\times \vec{\mathcal{A}} \right),\end{eqnarray} \tag{ 10 }$$

with Berry connection ${{\mathcal{A}}_{\mu }}={{\sum }_{n;{{E}_{n}}({\bf k})\lt 0}}\langle {{\Phi }_{n}}({\bf k})|{{\partial }_{{{k}_{\mu }}}}{{\Phi }_{n}}({\bf k})\rangle$ ($\mu =x,y$), where summation is taken for all occupied bands ${{E}_{n}}({\bf k})\lt 0$, and ${{\Phi }_{n}}({\bf k})$ is the wave-function of band n at momentum ${\bf k}$. Integrating the Berry curvature $\vec{\nabla }\times \vec{\mathcal{A}}$ shown in figure 2(b) over the Brillouin zone [33], we find that c = 1 for $\sqrt{{{(\mu +4{{t}_{0}})}^{2}}+{{\Delta }^{2}}}\lt {{V}_{z}}\lt \sqrt{{{\mu }^{2}}+{{\Delta }^{2}}}$ with $\mu \leqslant -2{{t}_{0}}$. The nonzero Chern number is attributed to the topologically nontrivial energy gap at Γ point $({{k}_{x}},{{k}_{y}})=(0,0)$ while those at $(0,-\pi )$, $(-\pi ,0)$ and $(-\pi ,-\pi )$ 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 $k\cdot p$ approximation around Γ point [34].

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 nm², which is divided into 400 × 200 square grids, corresponding to the Hamiltonian matrix of dimension ${{10}^{5}}\times {{10}^{5}}$ 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 ${{u}_{\uparrow }}=v_{\uparrow }^{*}$ and ${{u}_{\downarrow }}=v_{\downarrow }^{*}$ (displayed explicitly in figure 3(c) for the right hole), which results in ${{\gamma }^{\dagger }}=\gamma$, indicating that the two zero-energy states are Majorana states.

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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 ${{\Delta }^{2}}/\sqrt{{{\Delta }^{2}}+{{\mu }^{2}}}\sim \Delta$ with a small Fermi energy $\mu \sim \Delta$ [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 $\mu \gg \Delta$ and thus the minigap is small in order of ${{\Delta }^{2}}/\mu$.

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].

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 $t=2T$). 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.

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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 $t=2T\sim 4T$ 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 $t=4T\sim 6T$. 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.

In order to investigate the impact of position exchanging to MF states, we monitor the time evolution of MF wave-function $|\Psi (t)\rangle$ by solving the TDBdG equation numerically

$$\begin{eqnarray}&&{\rm i}\hbar \frac{{\rm d}}{{\rm d}t}|\Psi (t)\rangle =H(t)|\Psi (t)\rangle ,\end{eqnarray} \tag{ 11 }$$

where H(t) is given in (3) and depends on time in terms of the hopping rates ${{t}_{{\bf ij}}}$ and ${{t}_{\alpha {\bf i}}}$ 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 ${{10}^{6}}\times {{10}^{6}}$ for each time instant. Fortunately, it has been revealed that when the exponential operator is expanded by the Chebyshev polynomial

$$\begin{eqnarray}&&{\rm exp} [-{\rm i}H(t)\delta t/\hbar ]=\mathop{\sum }\limits_{n}{{c}_{n}}(\delta t){{T}_{n}}(H),\end{eqnarray} \tag{ 12 }$$

the coefficient ${{c}_{n}}(\delta t)$ decreases with n exponentially fast for small $\delta t$ [37]. Therefore, only several leading terms c_n are necessary for a sufficiently accurate estimate of ${\rm exp} [-{\rm i}H(t)\delta t/\hbar ]$. Moreover, the Chebyshev polynomials can be constructed based on the recursive relation ${{T}_{n}}(H)=2H{{T}_{n-1}}(H)-{{T}_{n-2}}(H)$, 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 $|\Psi (t+\delta t)\rangle \simeq {\rm exp} [-{\rm i}H(t)\delta t/\hbar ]|\Psi (t)\rangle$.

The wave function at time t = 0 is defined as $|\Psi (t=0)\rangle =|{{\Psi }_{{\rm L}}}(0)\rangle +|{{\Psi }_{{\rm R}}}(0)\rangle$ with $|{{\Psi }_{{\rm L}}}(0)\rangle$ and $|{{\Psi }_{{\rm R}}}(0)\rangle$ the zero-energy states at left and right holes, respectively, as shown in figures 4(a) and (b). We evaluate the projections of $|\Psi (t)\rangle$ onto the initial states ${{O}_{{\rm L}}}(t)=\langle {{\Psi }_{{\rm L}}}(0)|\Psi (t)\rangle$ and ${{O}_{{\rm R}}}(t)=\langle {{\Psi }_{{\rm R}}}(0)|\Psi (t)\rangle$ during adiabatical braiding processes and display them in figure 5 for $0\leqslant t\leqslant 6T$. Since four spinor components of MF satisfy ${{u}_{\uparrow }}=v_{\uparrow }^{*}$ and ${{u}_{\downarrow }}=v_{\downarrow }^{*}$ (see figure 3(c)), MF wave function can be simplified to $\Psi (\vec{r})={{[{{u}_{\uparrow }}(\vec{r}),{{u}_{\downarrow }}(\vec{r}),u_{\downarrow }^{*}(\vec{r}),-u_{\uparrow }^{*}(\vec{r})]}^{T}}$. The projections O_L and O_R then must be real numbers. The two MFs pick up opposite signs after exchanging their positions, which can be summarized by

$$\begin{eqnarray}&&{{\gamma }_{{\rm L}}}\to {{\gamma }_{{\rm R}}},\quad {{\gamma }_{{\rm R}}}\to -{{\gamma }_{{\rm L}}}.\end{eqnarray} \tag{ 13 }$$

Equation (13) can be written as ${{\gamma }_{{\rm L}/{\rm R}}}\to {{U}^{-1}}{{\gamma }_{{\rm L}/{\rm R}}}U$ with the unitary matrix $U={\rm exp} (\pi {{\gamma }_{{\rm R}}}{{\gamma }_{{\rm L}}}/4)$. This indicates that the braiding of MFs satisfies non-Abelian statistics [24, 25, 31, 40, 41, 43, 44], as will be shown explicitly below.

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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 [28–30]

$$\begin{eqnarray}\begin{array}{rcl} H & = & \int {{{\rm d}}^{2}}{\bf r}\left[ {{c}^{\dagger }}\left( -\frac{{{\nabla }^{2}}}{2m}-{{\varepsilon }_{F}} \right)c \right. \\ {} & {} & +\left. {{c}^{\dagger }}\left[ {{{\rm e}}^{{\rm i}\theta }}|\Delta ({\bf r})|({{k}_{x}}+i{{k}_{y}}) \right]{{c}^{\dagger }}+{\rm h}.{\rm C}. \right], \\ \end{array}\end{eqnarray} \tag{ 14 }$$

where the first term describes kinetic energy and chemical potential $-{{\varepsilon }_{F}}$, ${{c}^{\dagger }}$ is electron creation operator, and the second term gives pairing function $\Delta ({\bf r})$ with superconducting phase θ, k_x and k_y are electron momenta along x- and y-axis respectively. The Bogoliubov quasi-particle operator is ${{\gamma }^{\dagger }}=u{{c}^{\dagger }}+\nu c$, which satisfies the BdG equation

$$\begin{eqnarray}&&[H,{{\gamma }^{\dagger }}]=E{{\gamma }^{\dagger }}\end{eqnarray} \tag{ 15 }$$

with E the eigenenergy of quasi-particle. By taking Hermitian conjugation † at both sides of (15), we have

$$\begin{eqnarray}&&[H,\gamma ]=-E\gamma ,\end{eqnarray} \tag{ 16 }$$

indicating the particle-hole symmetry of BdG equation, i.e. ${{\gamma }^{\dagger }}(E)=\gamma (-E)$. Thus, the zero energy states must be double-degenerate, satisfying self-adjoint condition ${{\gamma }^{\dagger }}(E=0)=\gamma (E=0)$, 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: ${{\tilde{c}}^{\dagger }}={{{\rm e}}^{{\rm i}\theta /2}}{{c}^{\dagger }}$ and $\tilde{c}={{{\rm e}}^{-{\rm i}\theta /2}}c$, the new Bogoliubov quasi-particle operator is

$$\begin{eqnarray}&&{{\tilde{\gamma }}^{\dagger }}=u{{{\rm e}}^{{\rm i}\theta /2}}{{c}^{\dagger }}+\nu {{{\rm e}}^{-{\rm i}\theta /2}}c,\end{eqnarray} \tag{ 17 }$$

which can diagonalize the gauge-fixed Hamiltonian

$$\begin{eqnarray*}\begin{array}{rcl} \tilde{H} & = & \int {{{\rm d}}^{2}}{\bf r}\left[ {{\tilde{c}}^{\dagger }}\left( -{{\nabla }^{2}}/2m-{{\varepsilon }_{F}} \right)\tilde{c} \right. \\ {} & {} & +\left. {{\tilde{c}}^{\dagger }}\left[ |\Delta ({\bf r})|({{k}_{x}}+{\rm i}{{k}_{y}}) \right]{{\tilde{c}}^{\dagger }}+{\rm h}.{\rm C}. \right]. \\ \end{array}\end{eqnarray*}$$

In the case of $2\pi$ phase changing $\theta \to \theta +2\pi$, we have ${{\tilde{\gamma }}^{\dagger }}=-{{\gamma }^{\dagger }}$, which is an important result that can be employed to achieve non-Abelian statistics.

In a 2D spinless $p+{\rm i}p$ 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 $2\pi$. It is then easy to see that the Majorana wave-function bound to a vortex core picks up a sign $\gamma \to -\gamma$ after crossing the cut. By interchanging positions of vortices i and $i+1$ (see red arrows in figure 6), one observes that MF ${{\gamma }_{i}}$ passes through the cut of vortex $i+1$ and picks up $2\pi$ phase (see figure 6), which gives

$$\begin{eqnarray}&&{{\gamma }_{i}}\to -{{\gamma }_{i+1}}.\end{eqnarray} \tag{ 18 }$$

The other Majorana state at vortex $i+1$ does not cross the cut of vortex i, thus there is no sign change of MF wave-function

$$\begin{eqnarray}&&{{\gamma }_{i+1}}\to {{\gamma }_{i}}.\end{eqnarray} \tag{ 19 }$$

Therefore, we obtain the following rules for MFs evolution after exchanging positions of vortices hosting them

$$\begin{eqnarray}&&{{T}_{i}}:\left\{ \begin{array}{ccccccccccccccc} {{\gamma }_{i}}\to -{{\gamma }_{i+1}}, \\ {{\gamma }_{i+1}}\to {{\gamma }_{i}} \\ \end{array} \right.\end{eqnarray} \tag{ 20 }$$

with T_i the braiding operation, consistent with our results of braiding MFs (13).

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The unitary operator $\tau ({{T}_{i}})$ obeying ${{T}_{i}}{{\gamma }_{i}}=\tau {{({{T}_{i}})}^{-1}}{{\gamma }_{i}}\tau ({{T}_{i}})$ with T_i defined in (20) is given by [28]

$$\begin{eqnarray}&&\tau ({{T}_{i}})=\frac{1}{\sqrt{2}}\left( 1+{{\gamma }_{i+1}}{{\gamma }_{i}} \right),\end{eqnarray} \tag{ 21 }$$

implying non-Abelian statistics. To demonstrate this, we consider four vortices hosting four MFs ${{\gamma }_{1}},{{\gamma }_{2}},{{\gamma }_{3}}$ and ${{\gamma }_{4}}$, which combine into two complex fermions ${{\Psi }_{1}}=({{\gamma }_{1}}+{\rm i}{{\gamma }_{2}})/2$ and ${{\Psi }_{2}}=({{\gamma }_{3}}+{\rm i}{{\gamma }_{4}})/2$. The unitary operators describing two ways of braiding MFs on basis $[\mid 0\rangle ,\Psi _{1}^{\dagger }\mid 0\rangle ,\Psi _{2}^{\dagger }\mid 0\rangle ,\Psi _{1}^{\dagger }\Psi _{2}^{\dagger }\mid 0\rangle ]$ (${\mkern 1mu} \mid 0\rangle$ is the empty state) are [28]

$$\begin{eqnarray}\begin{array}{rcl} \tau ({{T}_{1}}) & = & \frac{1}{\sqrt{2}}\left( 1+{{\gamma }_{2}}{{\gamma }_{1}} \right)=\frac{1}{\sqrt{2}}\left[ 1+{\rm i}(\Psi _{1}^{\dagger }-{{\Psi }_{1}})(\Psi _{1}^{\dagger }+{{\Psi }_{1}}) \right] \\ {} & = & {{{\rm e}}^{-{\rm i}\pi /4}}\left( \begin{array}{ccccccccccccccc} 1 & 0 & 0 & 0 \\ 0 & {\rm i} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & {\rm i} \\ \end{array} \right), \\ \end{array}\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}\begin{array}{rcl} \tau ({{T}_{2}}) & = & \frac{1}{\sqrt{2}}\left( 1+{{\gamma }_{3}}{{\gamma }_{2}} \right)=\frac{1}{\sqrt{2}}\left[ 1+{\rm i}(\Psi _{2}^{\dagger }+{{\Psi }_{2}})(\Psi _{1}^{\dagger }-{{\Psi }_{1}}) \right] \\ {} & = & \frac{1}{\sqrt{2}}\left( \begin{array}{ccccccccccccccc} 1 & 0 & 0 & -{\rm i} \\ 0 & 1 & -{\rm i} & 0 \\ 0 & -{\rm i} & 1 & 0 \\ -{\rm i} & 0 & 0 & 1 \\ \end{array} \right). \\ \end{array}\end{eqnarray} \tag{ 23 }$$

The non-Abelian statistics is then verified by the noncommutative relation of unitary operators $\tau ({{T}_{1}})\tau ({{T}_{2}})\ne \tau ({{T}_{2}})\tau ({{T}_{1}})$ [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.

A toy model for realizing MFs in a N-site 1D spinless p-wave SC was proposed by Kitaev [8]

$$\begin{eqnarray}&&H=-\mu \mathop{\sum }\limits_{i=1}^{N}c_{i}^{\dagger }{{c}_{i}}-\mathop{\sum }\limits_{i=1}^{N-1}\left[ tc_{i}^{\dagger }{{c}_{i+1}}+|\Delta |{{{\rm e}}^{{\rm i}\theta }}{{c}_{i}}{{c}_{i+1}}+{\rm h}.{\rm C}. \right],\end{eqnarray} \tag{ 24 }$$

with μ the chemical potential, t nearest neighbor hopping rate, Δ superconducting gap function with phase θ. In the special case $\mu =0$ and $t=|\Delta |$, Hamiltonian (24) reduces to [8]

$$\begin{eqnarray}&&H=-{\rm i}t\mathop{\sum }\limits_{i=1}^{N-1}{{\gamma }_{B,i}}{{\gamma }_{A,{\rm i}+1}}\end{eqnarray} \tag{ 25 }$$

with

$$\begin{eqnarray}&&{{\gamma }_{A,i}}=-{\rm i}{{{\rm e}}^{-{\rm i}\theta /2}}c_{i}^{\dagger }+{\rm i}{{{\rm e}}^{{\rm i}\theta /2}}{{c}_{i}},\quad {{\gamma }_{B,i}}={{{\rm e}}^{-{\rm i}\theta /2}}c_{i}^{\dagger }+{{{\rm e}}^{{\rm i}\theta /2}}{{c}_{i}},\end{eqnarray} \tag{ 26 }$$

which are Majorana operators satisfying the self-adjoint condition $\gamma _{\alpha ,i}^{\dagger }={{\gamma }_{\alpha ,i}}$, $i=1,\cdots ,N$ is the lattice site index of the 1D system. Observing that the two terms ${{\gamma }_{A,1}}$ and ${{\gamma }_{B,N}}$ do not appear in the Hamiltonian (25), it can be concluded that the system can support two zero-energy quasi-particle states localized at two ends of the 1D SC (site 1 and N), while quasi-particles at other sites bind into complex fermions with finite energies. The two end states are just MFs, which appear in the topological phase for $|\mu |\lt t$ and disappear in the non-topological phase with $|\mu |\gt t$ [8, 42].

In 2D spinless $p+{\rm i}p$ SCs, MFs bound to vortex cores catch nonzero Berry phases while moving around other vortices, which give rise to non-Abelian statistics by exchanging vortices [28]. At first glance, it seemed impossible to realize non-Abelian statistics by exchanging Majorana states in 1D systems since vortices are absent, and two MFs collide with each other and fuse into finite energy states while exchanging them. Alicea et al proposed to interchange MFs in a T-junction nanowire network (see figure 7) and proved that the exchanging of MFs follows non-Abelian statistics [41]. They realize Kitaevʼs toy model by putting a SM nanowire with strong spin–orbit coupling on top of an s-wave SC [20, 21, 27, 42]. For simplicity, we adopt the Kitaev model to demonstrate braiding processes in the following.

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The authors consider two segments of 1D nanowires forming a T-junction [41], as shown in figure 7. Nanowire in the topological phase (green line) can be tuned into non-topological (light green line) region by adjusting local chemical potential μ using local gate voltage. To exchange positions of MFs in the T-junction (see figure 7), the authors first initialize the horizontal nanowire into topological state [41], hosting MFs ${{\gamma }_{A,1}}$ and ${{\gamma }_{B,N}}$ at the two ends (see figure 7(a)). The vertical nanowire is in the non-topological phase. By driving left part of horizontal nanowire into non-topological state and then vertical one into topological state, ${{\gamma }_{A,1}}$ is transported to the bottom of vertical nanowire (see figure 7(b)). Next, ${{\gamma }_{B,N}}$ at the right end is moved to the left end of horizontal nanowire (see figure 7(c)). Finally, ${{\gamma }_{A,1}}$ is moved to the right end of the horizontal nanowire (see figure 7(d)), which finishes interchanging of MFs. To track how ${{\gamma }_{A,1}}$ and ${{\gamma }_{B,N}}$ evolve after braiding, one can evaluate Berry phases acquired by the groundstate wave-function in principle. Here we adopt an alternative way to understand the statistics followed by interchanging MFs.

The topological energy gap of the 1D system remains nonzero while manipulating two MFs, which gives us the freedom to perform unitary transformations for Hamiltonian (24). For simplicity, we put Hamiltonian (24) to purely real while manipulating ${{\gamma }_{A,1}}$ and ${{\gamma }_{B,N}}$, which is only possible when superconducting phase θ is either 0 or π. The phase of the pairing function depends on the direction of site index i assignment since $|\Delta |{{{\rm e}}^{{\rm i}\theta }}{{c}_{i}}{{c}_{i+1}}=|\Delta |{{{\rm e}}^{{\rm i}(\theta +\pi )}}{{c}_{i+1}}{{c}_{i}}$. We denote $\theta =0$ by rightward/upward arrows, then $\theta =\pi$ must be represented by leftward/downward arrows (see figure 7). After exchanging two MFs, we end up with reversed arrows along the topological nanowire $\theta \to \theta +\pi$, indicating the sign flipping of uniform superconducting pairing Δ. To restore Hamiltonian back to its original form, we multiply a phase factor ${{{\rm e}}^{{\rm i}\pi /2}}$ to fermionic creation operator ${{f}^{\dagger }}=({{\gamma }_{A,1}}-i{{\gamma }_{B,N}})/2$ where θ is absorbed into

$$\begin{eqnarray}&&{{f}^{\dagger }}=({{\gamma }_{A,1}}-i{{\gamma }_{B,N}})/2\to {{{\rm e}}^{{\rm i}\pi /2}}{{f}^{\dagger }}=({{\gamma }_{B,N}}+i{{\gamma }_{A,1}})/2.\end{eqnarray} \tag{ 27 }$$

The above transformation gives

$$\begin{eqnarray}&&{{\gamma }_{A,1}}\to {{\gamma }_{B,N}},\quad {{\gamma }_{B,N}}\to -{{\gamma }_{A,1}},\end{eqnarray} \tag{ 28 }$$

which generates the non-Abelian statistics same as braiding of vortices in 2D spinless $p+{\rm i}p$ SCs as discussed above [28].

There are other possible schemes for realizing non-Abelian statistics in 1D nanowire system [24, 26, 43, 44]. Sau et al [24] proposed to transport MFs by tuning couplings between MFs, instead of driving MFs all the way along the T-junction by tuning chemical potential largely [41]. The Hamiltonian describing couplings between two MFs is

$$\begin{eqnarray}&&{{H}_{c}}={\rm i}\epsilon {{\gamma }_{1}}{{\gamma }_{2}},\end{eqnarray} \tag{ 29 }$$

which can form a complex fermion ${{f}^{\dagger }}=({{\gamma }_{1}}-i{{\gamma }_{2}})/2$ with finite energy. The number operator of fermion $n={{f}^{\dagger }}f$ has double-degenerated groundstates $|0\rangle$ and $|1\rangle$ for $\epsilon \to 0$. An energy gap is created between eigenstates $|0\rangle$ and $|1\rangle$ for finite . 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 ${{\gamma }_{2}}$, 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, ${{\gamma }_{2}}$ is moved to position C2 (see figure 8(b)). Next, ${{\gamma }_{1}}$ at A2 is transported to B1 (see figure 8(c)). Finally, ${{\gamma }_{2}}$ is moved to position A2 to complete the interchanging of MFs (see figure 8(d)). The trajectories followed by ${{\gamma }_{1}}$ and ${{\gamma }_{2}}$ are

$$\begin{eqnarray}\begin{array}{rcl} {{\gamma }_{1}} & : & {\rm A}2\to {\rm C}1\to {\rm B}1, \\ {{\gamma }_{2}} & : & {\rm B}1\to {\rm C}1\to {\rm C}2\to {\rm C}1\to {\rm A}2. \\ \end{array}\end{eqnarray} \tag{ 30 }$$

It is proven that interchanging of ${{\gamma }_{1}}$ and ${{\gamma }_{2}}$ in this way follows non-Abelian statistics since either ${{\gamma }_{1}}$ or ${{\gamma }_{2}}$ acquires a minus sign after exchanging positions [24].

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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 T-junctions depend on details of the junction, which may be difficult to control [24, 38]. As for the scheme by Sau et al, the coupling 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.