INTRODUCTION

Topologically-protected quantum manipulations with Majorana zero modes (MZM’s) are extensively studied theoretically and experimentally due to their exotic properties, including exchange statistics, and their possible use in platforms for topological quantum computation [1–3]. In particular, hybrid superconductor-topological insulator structures were discussed. Fu and Kane analyzed a topological Josephson junction between superconductor films on top of a topological insulator [4, 5] and demonstrated the appearance of Majorana edge states. Here we consider a setup where Majorana bound states are point-like structures bound to Josephson vortices in an external magnetic field perpendicular to the surface [6]. Such devices were discussed as a platform for topological quantum computation with the possibility of manipulation and braiding of Majorana bound states [7]. We analyze the tunnel coupling between the MZM’s and find that this coupling vanishes at zero chemical potential. This should be taken into account in the design of experiments with MZM’s on Josephson vortices and also suggests that coupling and hybridization of various MZM’s may be controlled, in particular, via the chemical potential. Note similar observations for a 2D vortex lattice [8].

MODEL HAMILTONIAN

We consider an S–TI–S Josephson junction between two s-wave superconducting (S) electrodes on top of a topological-insulator (TI) material, Fig. 1. Due to proximity effect, superconducting correlations are induced in the surface layer of the topological insulator. The states in this layer can be described by the Bogoliubov–de-Gennes (BdG) Hamiltonian \(H = \frac{1}{2}\int dxdy{\kern 1pt} {{\Psi }^{\dag }}h\Psi \), where \(\Psi = [{{\psi }_{ \uparrow }},{{\psi }_{ \downarrow }},\psi _{ \downarrow }^{\dag }, - \psi _{ \uparrow }^{\dag }{{]}^{T}}\).

Fig. 1.
figure 1

(Color online) S–TI–S Josephson junction in a transverse magnetic field \(\parallel \hat {z}\). Blue and orange spots indicate location of Majorana bound states.

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The single particle BdG Hamiltonian reads

$$\begin{gathered} h = v\left( {{{\sigma }_{x}}\left[ { - i\hbar {{\partial }_{x}}} \right] + {{\sigma }_{y}}\left[ { - i\hbar {{\partial }_{y}} + \frac{e}{c}{{A}_{y}}(x){{\tau }_{z}}} \right]} \right){{\tau }_{z}} \\ + \;\Delta (x,y){{\tau }_{ + }} + \Delta {\text{*}}(x,y){{\tau }_{ - }} - \mu {{\tau }_{z}}, \\ \end{gathered} $$
(1)

where the Pauli matrices σ and τ refer to the spin degree of freedom and to the Bogoliubov–Nambu particle–hole space (\({{\tau }_{ \pm }} = ({{\tau }_{x}} \pm i{{\tau }_{y}}){\text{/}}2\)), respectively.

The vector potential A describes the distribution of the transverse magnetic field \(\mathbf{H} = H\hat {z}\) around the gap of width W  between the superconducting electrodes:

$$H = \left\{ {\begin{array}{*{20}{c}} {{{H}_{0}}\exp \left[ { - \frac{{{\text{|}}x{\text{|}} - W{\text{/}}2}}{{{{\lambda }_{L}}}}} \right],}&{{\text{|}}x{\text{|}} > W{\text{/}}2{\kern 1pt} ,} \\ {{{H}_{0}},}&{ - W{\text{/}}2 < x < W{\text{/}}2.} \end{array}} \right.$$
(2)

We choose the gauge with \(\Delta = {{\Delta }_{0}}{{e}^{{i\phi (y)/2}}}\) in one superconductor, \(x > W{\text{/}}2\), and \(\Delta = {{\Delta }_{0}}{{e}^{{ - i\phi (y)/2}}}\) for \(x < - W{\text{/}}2\), while \(\Delta = 0\) in the gap \({\text{|}}x{\text{|}} < W{\text{/}}2\). The vector potential in this gauge has direction \(y\) and is given by:

$${{A}_{y}}(x) = \left\{ {\begin{array}{*{20}{c}} { - {{H}_{0}}x,}&{{\text{|}}x{\text{|}} < W{\text{/}}2,} \\ {{{\lambda }_{L}}{{H}_{0}}{{e}^{{ - \frac{{x - W/2}}{{{{\lambda }_{L}}}}}}} - \frac{{2{{\lambda }_{L}} + W}}{2}{\kern 1pt} {{H}_{0}},}&{x > W{\text{/}}2,} \\ { - {{A}_{y}}( - x){\kern 1pt} ,}&{x < 0.} \end{array}} \right.$$
(3)

The phase is chosen as \(\phi (y) = (2{{\lambda }_{L}} + W){{H}_{0}}\frac{{2\pi }}{{{{\Phi }_{0}}}}y\) with \({{\Phi }_{0}} = hc{\text{/}}2e\) being the superconducting flux quantum. We assume a not too strong field so that the corresponding magnetic length \({{l}_{B}} = [(2{{\lambda }_{L}} + W){{H}_{0}}{\text{/}}{{\Phi }_{0}}{{]}^{{ - 1}}}\) exceeds the relevant coherence length \(\xi = \hbar v{\text{/}}{{\Delta }_{0}}\), \({{l}_{B}} \gg \xi \).

Although in a realistic setup the system may be in the Pearl regime [9] with two-dimensional magnetostatics, we consider here the London regime [10]. While realistic situations may require further analysis, we expect on symmetry grounds that its basic properties will be the same.

SYMMETRIES

Properties of the solutions are to large extent determined by symmetries of the Hamiltonian, which we discuss here. First, as usual for the BdG Hamiltonian, the charge conjugation or particle–hole symmetry \(C = {{\sigma }_{y}}{{\tau }_{y}}K\) inverts energies: \({{C}^{{ - 1}}}hC = - h\).

Further, for \(\mu = 0\), the case of our special interest below, in addition there is a (quasi) time-reversal symmetry \(T = {{\sigma }_{x}}{{\tau }_{x}}K\), which commutes with the Hamiltonian, \({{T}^{{ - 1}}}hT = h\), and \({{T}^{2}} = 1\). Thus, for \(\mu = 0\), as usual, the product of C and T is the chiral symmetry \(S = {{\sigma }_{z}}{{\tau }_{z}}\). This set of symmetries describes the BDI symmetry class [11, 12].

Finally, we note that there is an extra symmetry operator,

$$F = {{\sigma }_{x}}{{\tau }_{x}}{{I}_{x}},$$
(4)

where the operator of \(x\) inversion acts as \({{I}_{x}}\psi (x,y) = \psi ( - x,y)\), \(I_{x}^{{ - 1}}{{A}_{y}}(x){{I}_{x}} = {{A}_{y}}( - x)\).

ONE-DIMENSIONAL SLICES

We split the Hamiltonian into \(h = {{h}_{0}} + {{h}_{1}}\), where:

$${{h}_{0}}(y) = - i\hbar v{\kern 1pt} {{\sigma }_{x}}{{\tau }_{z}}{{\partial }_{x}} + [\Delta (x,y){{\tau }_{ + }} + {\text{h}}{\text{.c}}.] - \mu {{\tau }_{z}}$$
(5)

and

$${{h}_{1}} = v{{\sigma }_{y}}\left[ { - i\hbar {{\partial }_{y}} + \frac{e}{c}{{A}_{y}}(x){{\tau }_{z}}} \right]{{\tau }_{z}}.$$
(6)

The Hamiltonian \({{h}_{0}}\) depends on y as a parameter via Δ since its phase ϕ is proportional to y. Hence, we can diagonalize it separately for each y and then take into account \({{h}_{1}}\), which glues these 1D states into 2D wavefunctions. Thus, we look for eigensolutions of:

$${{\hat {h}}_{0}}(y){\text{|}}\nu {{\rangle }_{y}} = {{\epsilon }_{\nu }}(y){\text{|}}\nu {{\rangle }_{y}}$$
(7)

or equivalently, \({{\hat {h}}_{0}}(\phi ){\text{|}}\nu {{\rangle }_{\phi }} = {{\epsilon }_{\nu }}(\phi ){\text{|}}\nu {{\rangle }_{\phi }}\), where y is a parameter and \({\text{|}}\nu {{\rangle }_{y}}\) represents a wavefunction \({{f}_{{\nu y}}}(x)\).

The resulting energies depend on system parameters, with examples shown in Figs. 2 and 3. The eigenstates can be classified by the eigenvalues ±1 of the symmetry operator F (as indicated with the blue, \(F = 1\), and orange, \(F = - 1\), color in the figures) and \({{\sigma }_{x}}\) (a symmetry of \({{h}_{0}}\), see below).

Fig. 2.
figure 2

(Color online) Eigenenergies \({{\epsilon }_{n}}(\phi )\) of \({{h}_{0}}\) for \(W = 2.5\xi \).

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Fig. 3.
figure 3

(Color online) Eigenenergies \({{\epsilon }_{n}}(\phi )\) of \({{h}_{0}}\) for \(W = 6.0\xi \).

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The spectrum \({{\epsilon }_{\nu }}\) depends on ϕ (or y), and has zero modes [4] at \(\phi = \pi + 2\pi n\): here the gap is purely imaginary \(\Delta (x) = i{{( - 1)}^{n}}{{\Delta }_{0}}{\text{sgn}}{\kern 1pt} x\), and

$${{h}_{0}} = - iv{{\sigma }_{x}}{{\tau }_{z}}{{\partial }_{x}} - {{( - 1)}^{n}}{{\Delta }_{0}}{{\tau }_{y}}{\text{sgn}}{\kern 1pt} x$$
(8)

has two zero modes, which are eigenstates of F = \({{\sigma }_{x}}{{\tau }_{x}}{{I}_{x}}\) with \(F = ( - {{1)}^{n}}\) (\(F = 1\) at \(\phi = \ldots - 3\pi ,\pi ,5\pi \ldots \) and \(F = - 1\) at \(\phi = \ldots - \pi ,3\pi ,7\pi \ldots \)):

$$\left( {\begin{array}{*{20}{c}} 1 \\ {{{\sigma }_{x}}} \\ {{{\sigma }_{x}}F} \\ F \end{array}} \right){{e}^{{ - \int_0^{|x|} {|\Delta (x')|dx'} }}}$$
(9)

(note another gauge used in [4]).

It is convenient to mark an eigenstate \({{\epsilon }_{\nu }}(y)\) of \({{h}_{0}}\) with the sign of its slope (\( \pm = - {{\sigma }_{x}}\)) and an integer \(n\), where its energy crosses zero, \({{\epsilon }_{\nu }}(\phi = \pi + 2\pi n) = 0\), so that \(\nu = (n, \pm )\).

We observe that the spectrum is \(2\pi \)-periodic, \({{\epsilon }_{\nu }}(\phi + 2\pi ) = {{\epsilon }_{{\nu '}}}(\phi )\), however the wavefunctions are \(4\pi \)‑periodic, \({{f}_{\nu }}(\phi + 4\pi ) = {{f}_{{\nu ''}}}(\phi )\). This is another manifestation of the symmetry F.

CONSTRUCTION OF 2D WAVE FUNCTIONS

Having obtained the eigenvectors \({\text{|}}\nu {{\rangle }_{y}}\) (7) of \({{h}_{0}}\) for each y, we use them as a basis to construct solutions

$$\psi (x,y) = \sum\limits_\nu {{\alpha }_{\nu }}(y){{f}_{{\nu y}}}(x),$$
(10)

of the 2D BdG equation:

$$({{h}_{0}} + {{h}_{1}})\psi = E\psi .$$
(11)

Rewriting the BdG equation as \((E - {{h}_{0}})\psi = {{h}_{1}}\psi \), we find:

$$(E - {{\epsilon }_{\lambda }}(y)){{\alpha }_{\lambda }}(y) = \sum\limits_\nu (h_{{{\text{kin}}}}^{{\lambda \nu }}{{\partial }_{y}} + \tilde {h}_{1}^{{\lambda \nu }}){{\alpha }_{\nu }}(y),$$
(12)
$$h_{{{\text{kin}}}}^{{^{{\lambda \nu }}}}{{( - i\hbar v)}_{y}}{{\langle \lambda {\text{|}}{{\sigma }_{y}}{{\tau }_{z}}{\text{|}}\nu \rangle }_{y}},$$
(13)
$$\tilde {h}_{1}^{{\lambda \nu }} = ( - i\hbar v){{{\kern 1pt} }_{y}}{{\langle \lambda {\text{|}}{{\sigma }_{y}}{{\tau }_{z}}\left( {{{\partial }_{y}} + \frac{{ie}}{{\hbar c}}{{A}_{y}}(x){{\tau }_{z}}} \right){\text{|}}\nu \rangle }_{y}}.$$
(14)

We note that Fu and Kane [4] used a different, y-independent basis \({\text{|}}\nu {{\rangle }_{{y = {{y}_{\lambda }}}}}\), suitable at \(y \approx {{y}_{\lambda }}\) for low-energy states near a node \({{\epsilon }_{\lambda }}({{y}_{\lambda }}) = 0\). Since we consider also ranges away from nodes, we chose \({\text{|}}\nu {{\rangle }_{y}}\) above. This brings a new term \({{\tilde {h}}_{1}}\) compared to [4], which we will treat perturbatively.

ADDITIONAL SYMMETRIES OF \({{h}_{0}}\) AT \(\mu = 0\)

The Hamiltonian \({{h}_{0}}\) has an extra symmetry:

$$[{{h}_{0}},{{\sigma }_{x}}] = 0,$$
(15)

which at \(\mu = 0\) allows one to use an alternative time reversal \(\Theta = {{\tau }_{x}}K = {{\sigma }_{x}}T\) and an alternative chiral symmetry \(P = {{\sigma }_{y}}{{\tau }_{z}} \propto {{\sigma }_{x}}S\). Note that Θ and P are not symmetries of the full Hamiltonian \({{h}_{0}} + {{h}_{1}}\): indeed, \(\{ {{h}_{1}},\Theta \} = 0\) whereas \([{{h}_{0}},\Theta ] = 0\), also \([{{h}_{1}},P] = 0\) whereas \(\{ {{h}_{0}},P\} = 0\).

The chiral symmetry P maps between \({\text{|}}n \pm \rangle \) and squares to unity, \({{P}^{2}} = 1\). Our choice (9) of the basis implies that \(P{\text{|}}n + \rangle = i{\text{|}}n - \rangle \) and \(P{\text{|}}n - \rangle = - i{\text{|}}n + \rangle \).

ZERO MODES

Let us first omit the \({{\tilde {h}}_{1}}\) term in Eq. (12). As the symmetries above show, \({{h}_{{{\text{kin}}}}}\) couples only the levels within each charge conjugate pair \({\text{|}}n \pm \rangle \), and Eq. (12) factorizes into 2 × 2 equations:

$$E\left( {\begin{array}{*{20}{c}} {{{\alpha }_{{n + }}}(y)} \\ {{{\alpha }_{{n - }}}(y)} \end{array}} \right) = {{h}_{{{\text{eff}}}}}\left( {\begin{array}{*{20}{c}} {{{\alpha }_{{n + }}}(y)} \\ {{{\alpha }_{{n - }}}(y)} \end{array}} \right),$$
(16)

where

$${{h}_{{{\text{eff}}}}} = - iv{{\rho }_{y}}{{\partial }_{y}} + {{\epsilon }_{n}}(y){{\rho }_{z}}.$$
(17)

Here, ρ are a new set of Pauli matrices in the basis \(n{\kern 1pt} \pm \).

The energy \({{\epsilon }_{n}}\) vanishes at coordinates \(y = {{y}_{n}}\) with phase \(\phi ({{y}_{n}}) = \pi + 2\pi n\). Around this point we linearize \({{\epsilon }_{n}}(y) = \alpha {\kern 1pt} (y - {{y}_{n}})\) and obtain an exactly solvable [6, 7]

$${{h}_{{{\text{eff}}}}} = - iv{{\rho }_{y}}{{\partial }_{y}} + \alpha {\kern 1pt} (y - {{y}_{n}}){{\rho }_{z}}$$
(18)

with \(\alpha \propto {{\partial }_{y}}\phi \). This equation has a zero mode localized near \({{y}_{n}}\) and a set of “Landau levels,” cf. Fig. 4. In the regime we consider, \({{l}_{B}} \gg \xi \), many Landau levels fit below the gap \({{\Delta }_{0}}\).

Fig. 4.
figure 4

(Color online) Schematic drawing of energy levels near a single node.

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COUPLING AND HYBRIDIZATION OF MAJORANA ZERO MODES

Taking into account the \({{\tilde {h}}_{1}}\) terms in Eq. (12) may tunnel-couple zero modes at different nodes \(\phi = \pi + 2\pi n\), pushing them away from zero energy. We show that this is not the case, and at \(\mu = 0\) the modes remain at zero energy, forming a flat band. This observation is similar to that of [8] for a 2D system with a vortex lattice.

Indeed, from Eq. (18) one observes that the zero mode at each node, \({\text{|}}n + \rangle + S{\text{|}}n - \rangle \), is chiral with the same chirality \(S = {{\sigma }_{z}}{{\tau }_{z}} = - {\text{sgn}}\alpha \) for all nodes. We show that adding \({{\tilde {h}}_{1}}\) does not alter this property.

We note first that effects of \({{\tilde {h}}_{1}}\) on the energies of the zero modes are weak and can be treated perturbatively, and hence the zero modes may acquire only a weak admixture of other, non-zero states. On the other hand, the chiral symmetry implies that all zero-energy solutions have full chirality, while all nonzero-energy solutions are equal-weight superpositions of two chiralities. Since without \({{\tilde {h}}_{1}}\) all zero modes have the same chiralities, a weak perturbation cannot push them away from zero.

For a sufficiently wide contact \(W \gg \xi \), the spectrum \({{\epsilon }_{\nu }}(y)\) is below \({{\Delta }_{0}}\) and linear in a wide range of phases, so that the zero mode wavefunctions can be found explicitly in the range of interest. Then one observes the vanishing tunnel coupling explicitly. The zero states, for instance, in two neighboring orange nodes \(\phi = - \pi \) and \(\phi = 3\pi \) are both \( \propto {\kern 1pt} {{(1,0,0,1)}^{T}}\), and the matrix element of the Hamiltonian between them vanishes, \(\langle {{\psi }_{{ - \pi }}}{\text{|}}h{\text{|}}{{\psi }_{{3\pi }}}\rangle = 0\) as follows from its matrix structure (and is a consequence of the fact that h changes chirality).

While the zero modes remain decoupled, other levels near a certain node can be perturbed due to coupling to other nodes. This is schematically indicated in Fig. 4 by a finite width of the corresponding Landau level.

Our result is in agreement with the general classification of zero modes [13], which implies that the number of MZM’s is topologically protected (note the same chirality of all MZM’s) and given by the total phase drop accumulated around the defects within the Josephson junction.

DISCUSSION

We note that coupling between the MZM’s in this setting was suggested as a method to braid them [7], which would allow for topologically-protected quantum operations [14, 15]. While the coupling vanishes at \(\mu = 0\), a finite coupling may be achieved for \(\mu \ne 0\) (in fact, reaching \(\mu = 0\) may be experimentally challenging), which also permits controlling its strength.

As we have discussed above, at low magnetic fields and large intervortex distance, Majorana zero modes are topologically protected [13]. Unlike the Andreev bound states in conventional Josephson junctions [16], which are sensitive to normal scattering and spin-orbit coupling, the MZM’s are protected due to topological helical nature of the surface states [6]. Potential disorder in the junction may affect the MZM’s, weak effect can be expected for correlation lengths below \({{l}_{B}}\), while influence of longer correlations on the MZM’s needs further analysis.

Furthermore, finite coupling can be realized if two MZM’s have different chiralities. Since the chirality depends on the sign of the magnetic field (sgn\({\kern 1pt} \alpha \) in Eq. (18)), this can be effected with a nonuniform magnetic field distribution: for instance, with a “domain wall” of \(H(y) = {{H}_{0}}{\text{sgn}}{\kern 1pt} y\) or an oscillatory \(H(y)\). The corresponding structures would allow for controlled quantum operations with Majorana modes.