Manifestation of Majorana modes overlap in the Aharonov–Bohm effect

One can treat the system under consideration as the normal (N) and superconducting (S) contacts coupled by the top (1) and bottom (2) arms, which are one-dimensional normal wires, as it is schematically depicted in figure 1. The model Hamiltonian has a following general form:

$$\begin{equation}\hat{H}={\hat{H}}_{N}+{\hat{H}}_{D}+{\hat{H}}_{t}.\end{equation} \tag{ 1 }$$

The summand ${\hat{H}}_{N}$ characterizes the single-band normal contact,

$$\begin{equation}{\hat{H}}_{N}=\sum\limits _{k\sigma }\left({\xi }_{k}-\frac{eV}{2}\right){c}_{k\sigma }^{+}{c}_{k\sigma },\end{equation} \tag{ 2 }$$

where c_kσ —an annihilation operator of the electron with momentum k, spin σ and energy ξ_k = ɛ_k − μ in the contact; μ—a chemical potential of the system; eV—an electric-field energy related to bias voltage. The second term is a tight-binding Hamiltonian which models a device consisting of two normal wires (or top and bottom arms) coupled to the opposite ends of the SC wire, ${\hat{H}}_{D}={\hat{H}}_{1}+{\hat{H}}_{2}+{\hat{H}}_{S}+{\hat{H}}_{\tau }$,

$$\begin{align}\hfill {\hat{H}}_{i}& ={\xi }_{i}\sum\limits _{\sigma ;j=1}^{{N}_{i}}{d}_{ij\sigma }^{+}{d}_{ij\sigma }\hfill \\ \hfill & \quad -\sum\limits _{j=1}^{{N}_{i}-1}\left(\frac{{t}_{i}}{2}{d}_{ij\sigma }^{+}{d}_{i,j+1,\sigma }+i\frac{{\alpha }_{i}^{x}}{2}{d}_{ij\sigma }^{+}{d}_{i,j+1,\bar{\sigma }}+\mathrm{H}.\mathrm{c}.\right),\hfill \end{align} \tag{ 3 }$$

$$\begin{align}\hfill {\hat{H}}_{S}& =\sum\limits _{j=1}^{N}\left[{\xi }_{S}\sum\limits _{\sigma }\,{a}_{j\sigma }^{+}{a}_{j\sigma }-h{a}_{j{\uparrow}}^{+}{a}_{j{\downarrow}}+{\Delta}\left({a}_{j{\uparrow}}{a}_{j{\downarrow}}+\mathrm{H}.\mathrm{c}.\right)\right]\hfill \\ \hfill & \quad -\sum\limits _{\sigma ;j=1}^{N-1}\left[\frac{t}{2}\,{a}_{j\sigma }^{+}{a}_{j+1,\sigma }+\frac{{\alpha }^{y}}{2}\sigma {a}_{j\sigma }^{+}{a}_{j+1,\bar{\sigma }}+\mathrm{H}.\mathrm{c}.\right],\hfill \end{align} \tag{ 4 }$$

$$\begin{align}\hfill {\hat{H}}_{\tau }& =-\sum\limits _{\sigma }\left(\frac{{\tau }_{1}}{2}\,{d}_{1,{N}_{1},\sigma }^{+}{a}_{1\sigma }+\frac{{\tau }_{2}}{2}{d}_{21\sigma }^{+}{a}_{N\sigma }\right.\hfill \\ \hfill & \quad \left.+i\frac{{\alpha }_{i}^{x}}{2}\,{d}_{1,{N}_{1},\sigma }^{+}{a}_{1\bar{\sigma }}-i\frac{{\alpha }_{i}^{x}}{2}{d}_{21\sigma }^{+}{a}_{N\bar{\sigma }}+\mathrm{H}.\mathrm{c}.\right),\hfill \end{align} \tag{ 5 }$$

where ${\xi }_{i}={\varepsilon }_{i}-\mu ,{t}_{i},{\alpha }_{i}^{x}$—a single-electron energy, hopping parameter and intensity of the Rashba spin–orbit interaction in the ith arm (i = 1, 2), respectively; ξ_S = ɛ − μ, t, α^y —the analogous quantities in the SC segment of the device; Δ—a strength of SC pairing induced by the proximity effect; τ_i —a parameter of hopping between the ith arm and SC wire; h—the Zeeman splitting energy caused by the longitudinal magnetic field, B (see figure 1). Note that to realize the topologically nontrivial phase and the Majorana modes, b₁ and b₂, the magnetic field has to be oriented perpendicular to the vector of effective Rashba field in the SC segment [1, 2]. The SC wire and ith arm contains N and N_i sites, respectively.

The last term in (1) takes into account tunneling processes between the normal contact and structure,

$$\begin{equation}{\hat{H}}_{t}=\sum\limits _{k\sigma }\,{c}_{k\sigma }^{+}\left({t}_{1}\,{\text{e}}^{-\text{i}\frac{\phi }{2}}{d}_{11\sigma }+{t}_{2}\,{\text{e}}^{\text{i}\frac{\phi }{2}}{d}_{2,{N}_{2},\sigma }\right)+\mathrm{H}.\mathrm{c}.,\end{equation} \tag{ 6 }$$

where t_1,2—intensities of tunnel couplings; ϕ = 2πΦ/Φ₀—the AB phase induced by a magnetic flux Φ penetrating the device; Φ₀ = h/e—the flux quantum.

To calculate a stationary charge current in the normal contact, $I=e\frac{\partial }{\partial t}{\sum }_{k\sigma }\langle {c}_{k\sigma }^{+}(t){c}_{k\sigma }(t)\rangle$, the Hamiltonian (1) is diagonalized based on the Gorkov–Nambu operators [37, 38], ${\hat{f}}_{l}={\left({f}_{l{\uparrow}},{f}_{l{\downarrow}}^{+},{f}_{l{\downarrow}},{f}_{l{\uparrow}}^{+}\right)}^{\mathrm{T}}$, where f_lσ —the second quantization operator of particle in one of the four above-described subsystems, l—an index of site or wave vector. Matrix nonequilibrium GFs [39] for the introduced field operators can be defined as ${\hat{G}}_{lm}^{ab}\left({\tau }_{a},{\tau }_{b}\right)=-i\left\langle {T}_{K}{\hat{f}}_{l}({\tau }_{a}){\hat{f}}_{m}^{+}({\tau }_{b})\right\rangle$, where T_K —an operator of time ordering on the Keldysh contour. The indices a, b = +, − of the GFs indicate that the time argument of the corresponding operator belongs to either lower or upper branch of the Keldysh contour.

Performing a series of transformations, the following expression for the current is obtained [40, 41]:

$$\begin{align}\hfill I& =\frac{2e}{h}\sum\limits _{i,j=1,2}\underset{-\infty {}}{\overset{+\infty }{\int }}\mathrm{d}\omega \,\mathrm{Tr}\left[\hat{\sigma }\,\mathrm{R}\mathrm{e}\left\{{\hat{{\Sigma}}}_{ij}^{r}\left(\omega \right){\hat{G}}_{ji}^{+-}\left(\omega \right)\right.\right.\hfill \\ \hfill & \quad \left.\left.+{\hat{{\Sigma}}}_{ij}^{+-}\left(\omega \right){\hat{G}}_{ji}^{a}\left(\omega \right)\right\}\right],\hfill \end{align} \tag{ 7 }$$

where $\hat{\sigma }=\mathrm{diag}\left(1,-1,1,-1\right)$. The matrix blocks of the device's advanced GF, ${\hat{G}}_{ji}^{a}$, are determined by the solution of the Dyson equation,

$$\begin{equation}{\hat{G}}^{a}={\left[{\left(\omega -{\hat{h}}_{D}-{\hat{{\Sigma}}}^{r}\left(\omega \right)\right)}^{-1}\right]}^{+},\end{equation} \tag{ 8 }$$

where ${\hat{{\Sigma}}}^{r}\left(\omega \right)$—a matrix of the retarded self-energy function reflecting the influence of the contact on the device. Assuming that the normal contact band is wide, one has the following nonzero blocks of ${\hat{{\Sigma}}}^{r}\left(\omega \right)$:

$$\begin{align}\hfill {\hat{{\Sigma}}}_{11\left(22\right)}^{r}& =-\frac{i}{2}{\hat{{\Gamma}}}_{11\left(22\right)},\hspace{25.0pt}{\hat{{\Sigma}}}_{21\left(12\right)}^{r}=-\frac{i}{2}{\hat{{\Gamma}}}_{12}{\hat{{\Phi}}}_{\pm }^{2},\hfill \\ \hfill {\hat{{\Phi}}}_{\pm }& =\mathrm{diag}\left({\mathrm{e}}^{\pm \mathrm{i}\frac{\phi }{2}},{\mathrm{e}}^{\mp \mathrm{i}\frac{\phi }{2}},{\mathrm{e}}^{\pm \mathrm{i}\frac{\phi }{2}},{\mathrm{e}}^{\mp \mathrm{i}\frac{\phi }{2}}\right),\hfill \end{align} \tag{ 9 }$$

where ${\hat{{\Gamma}}}_{ii}={{\Gamma}}_{i}{\hat{\tau }}_{0}$, ${{\Gamma}}_{i}=2\pi \rho {\left({t}_{i}/2\right)}^{2}$—functions characterizing the broadening of the device levels (i = 1, 2); ${\hat{\tau }}_{0}$—an 4 × 4 identity matrix; ρ—the contact density of states; ${{\Gamma}}_{12}=\sqrt{{{\Gamma}}_{1}{{\Gamma}}_{2}}$. The matrix lesser GF, ${\hat{G}}_{ij}^{+-}$, in (7) is found from the solution of the Keldysh equation, ${\hat{G}}^{+-}={\hat{G}}^{r}{\hat{{\Sigma}}}^{+-}{\hat{G}}^{a}$. Its nonzero blocks are

$$\begin{equation}{\hat{{\Sigma}}}_{ij}^{+-}=-2{\hat{{\Sigma}}}_{ij}^{r}\cdot \mathrm{diag}\left({n}_{h},{n}_{e},{n}_{h},{n}_{e}\right),\quad i,j=1,2,\end{equation} \tag{ 10 }$$

where ${n}_{e\left(h\right)}\equiv n\left(\omega \mp eV/2\right)$—the Fermi–Dirac distribution functions.

For simplicity, in the following numerical analysis we will assume that ξ₁ = ξ₂, t₁ = t₂ = t, N₁ = N₂. At the same time, to study the influence of spin–orbit coupling in the arms on the AB effect in more details the cases of identical and different normal wires will be analyzed separately, namely: (1) ${\alpha }_{1}^{x}=-{\alpha }_{2}^{x}={\alpha }^{x}$; (2) ${\alpha }_{1}^{x}=-2{\alpha }_{2}^{x}={\alpha }^{x}$. In order to model real InAs, InSb semiconducting wires of μm length the lattice constant is chosen as a = 50 nm. Then, using the effective mass value found experimentally, m = 0.015m₀, the hopping parameter is t = ℏ²/ma² ≈ 2 meV [6]. Throughout all the numerical calculations the other energy parameters are measured in the units of t. Namely, ɛ_1,2 = 0, ɛ = 1, μ = 0, Δ = 0.2, α^x = 0.4, α^y = 0.3, Γ_1,2 = 0.01, τ_1,2 = 0.1, k_B T ≈ 0, N₁ = 30 and N = 60.

To extract the effect related to the presence of SC segment in the device on low-energy interference transport we start with the limiting case when α^x = 0. The corresponding Zeeman-energy dependence of conductance is shown in figure 2(a). The behavior of $G\left(h\right)$ for the zero (dashed curve) and nonzero (solid curve) magnetic flux is qualitatively similar. At the low fields, h < Δ, the SC wire is in the trivial phase and the energy of the device lowest state, E₁, is nonzero (see dotted curve) leading to G = 0. In opposite, if h > Δ the phase is nontrivial. At the intermediate magnetic fields, close to the critical one, E₁ ≈ 0 resulting in the monotonous dependence of $G\left(h\right)$. At ϕ = π/2 the conductance reaches 4G₀ value implying that both Majorana modes are involved in the transport between the normal and superconducting contacts contributing 2G₀ each,

$$\begin{equation}{b}_{1}=\sum\limits _{\sigma ;j=1}^{N+2{N}_{1}}{w}_{j\sigma }{\gamma }_{j\sigma A},\hspace{25.0pt}{b}_{2}=\sum\limits _{\sigma ;j=1}^{N+2{N}_{1}}{z}_{j\sigma }{\gamma }_{j\sigma B},\end{equation} \tag{ 11 }$$

$$\begin{equation}{w}_{j\sigma }={v}_{j\sigma }^{\ast }+{u}_{j\sigma },\hspace{25.0pt}{z}_{j\sigma }=i\left({v}_{j\sigma }^{\ast }-{u}_{j\sigma }\right),\end{equation} \tag{ 12 }$$

where u_jσ , v_jσ —the spin-dependent Bogoliubov coefficients of the device lowest excitation; ${\gamma }_{j\sigma A\left(B\right)}={\gamma }_{j\sigma A\left(B\right)}^{+}$—the Majorana operators of type A(B). At the high fields the hybridization of Majorana wave functions becomes nonzero giving rise to the oscillations of ${E}_{1}\left(h\right)$ and $G\left(h\right)$.

Zoom In Zoom Out Reset image size

At zero AB phase the conductance is significantly suppressed. If the spin–orbit coupling in the arms is taken into account and $\vert {\alpha }_{1}^{x}\vert =\vert {\alpha }_{2}^{x}\vert$ that there is a slight quantitative difference from the α^x = 0 case (e.g., compare the solid curve in figure 2(a) with the dotted curve in figure 2(b)). When $\vert {\alpha }_{1}^{x}\vert \ne \vert {\alpha }_{2}^{x}\vert$ the divergence can be significant, which is especially prominent at ϕ = π/2 (see solid curve in figure 2(b)).

The AB oscillations of conductance are plotted in figure 3(a) when the device is in the nontrivial phase with the NOMMs. Both at α^x = 0 and at α^x ≠ 0, $\vert {\alpha }_{1}^{x}\vert =\vert {\alpha }_{2}^{x}\vert$ the peaks appear when $\phi =\pi \left(n+1/2\right)$, $n\in \mathbb{Z}$ (see dotted and dashed curves). In turn, the different spin–orbit interaction in the arms leads to their shift as shown by solid curve.

Zoom In Zoom Out Reset image size

In general, spin–orbit coupling in the system results in electron spin precession around the Rashba vector. Then, an additional phase carriers acquire in such media propagating from jth to j'th site [42] can be related to the change of the angle between the spins of the device lowest excitation, ψ₁, at these sites, ${\Theta}\left({\mathbf{S}}_{1}\left(j\right),{\mathbf{S}}_{1}\left({j}^{\prime }\right)\right)$, where

$$\begin{equation}{\mathbf{S}}_{1}\left(j\right)={\psi }_{1}^{+}\left(j\right)\boldsymbol{\sigma }\frac{{\tau }_{0}+{\tau }^{z}}{2}{\psi }_{1}\left(j\right).\end{equation} \tag{ 13 }$$

Here $\boldsymbol{\sigma }=\left({\sigma }^{x},{\sigma }^{y},{\sigma }^{z}\right),\enspace {\tau }^{z}$—the Pauli matrices acting in the spin and particle-hole space, respectively.

In the presence of mirror symmetry (α^x = 0 or α^x ≠ 0, $\vert {\alpha }_{1}^{x}\vert =\vert {\alpha }_{2}^{x}\vert$) the angles between the spins at the opposite ends of the SC section, ${\Theta}\left({\mathbf{S}}_{1}\left(j={N}_{1}+1\right),{\mathbf{S}}_{1}\left(j={N}_{1}+N\right)\right)\equiv {{\Theta}}_{S1,S2}$, and at the edges of the device, ${\Theta}\left({\mathbf{S}}_{1}\left(j=1\right),{\mathbf{S}}_{1}\left(j=2{N}_{1}+N\right)\right)\equiv {{\Theta}}_{1,2}$, are the same. The latter is clearly seen in figure 3(b) where a ratio Θ_1,2/Θ_S1,S2 as a function of α^x is displayed (see dashed line). On the contrary, if $\vert {\alpha }_{1}^{x}\vert \ne \vert {\alpha }_{2}^{x}\vert$ the angles match only for the certain values of α^x (see vertical dotted lines in figure 3(b)). Outside these points of the parametric space the conductance resonances appear at ϕ ≠ π(n + 1/2) (see solid curve in figure 3(a)).

One of the main results of this study is displayed in figure 4(a). As the Zeeman energy increases the positions of the maxima in the AB effect change and the oscillation period doubles. This process is accompanied by the increasing overlap of the Majorana wave functions. The corresponding probability densities, ${M}_{1}\left(j\right)={\sum }_{\sigma }\vert {w}_{j\sigma }{\vert }^{2}$ and ${M}_{2}\left(j\right)={\sum }_{\sigma }\vert {z}_{j\sigma }{\vert }^{2}$, are shown in figures 4(b) and (c) for $\vert {\alpha }_{1}^{x}\vert \ne \vert {\alpha }_{2}^{x}\vert$ at intermediate and high magnetic fields, respectively. To observe such a hybridization in transport the modes must have nonzero probability density at both ends of the device. In order for the leakage of Majorana modes [43] into the arms to be possible in our system, the normal wires have to be quantum wells, ɛ_1,2 ⩽ ɛ. Then, the desired situation of ${M}_{1}\left(N+2{N}_{1}\right),{M}_{2}\left(1\right)\ne 0$ realizes at the high magnetic fields as it can be seen from figure 4(d) where the h-dependencies of ${M}_{1}\left(1\right)$ and ${M}_{1}\left(N+2{N}_{1}\right)$ are plotted (the b₂ mode behaves identically). In the trivial phase (h < 0.2 or h > 2) the device lowest-energy excitation is localized in one of the arms since $\vert {\alpha }_{1}^{x}\vert \ne \vert {\alpha }_{2}^{x}\vert$ leading to either ${M}_{1}\left(1\right)=0$ or ${M}_{1}\left(N+2{N}_{1}\right)=0$. Immediately before the topological phase transition the ${M}_{1}\left(j\right)$ is mostly concentrated in the SC section causing its sudden drop at both edges of the device. In the nontrivial phase at the intermediate fields (0.2 < h ≲ 0.5) the ${M}_{1}\left(j\right)$ is localized predominantly at the boundary of the SC wire and arm 1 (see figure 4(b)). Then, at the high fields, 0.5 ≲ h < 2, the process of b₁ delocalization results in ${M}_{1}\left(N+2{N}_{1}\right)\ne 0$ at the expense of the ${M}_{1}\left(1\right)$ reduction albeit ${M}_{1}\left(N+2{N}_{1}\right)\ll {M}_{1}\left(1\right)$. In turn, it leads to the narrowing of conductance resonant peaks (e.g. see figures 2(b) and 4(a)).

Zoom In Zoom Out Reset image size

When the in-plane magnetic field increases the probability density in the central part of the SC wire becomes nonzero at the expense of decrease of the probability density at its ends and the arms.

Note that the described picture seems feasible in the modern experiments since to reach ballistic transport regime the short wires are used ($\sim 0.1-1\enspace \mu$m). Taking additionally into account the available proximity-induced SC gaps ($\sim 0.1-1$ meV) one can conclude that the Majorana modes are able to extend over the whole structure [11, 16]. In the next section considering a spinless inference system we will analytically show that the direct interaction of the mode ${b}_{1\left(2\right)}$ with the arm $2\left(1\right)$ induces the AB peaks shift.

The proposed setup can be effective in distinguishing between the nontopological quasi-Majorana state and nonlocal topological Majorana state with spatially separated Majorana modes. The former emerge in the trivial phase due to the presence of smooth electrostatic and/or SC pairing potentials around the N/S interfaces [22, 44, 45]. Despite the fact that in this case the overlap of the Majorana modes can vary significantly and be even comparable with the one in the nontrivial phase, they are still localized in such an inhomogeneous region. As a result, although the linear-response conductance can reach 4G₀ for this type of states [45] it is difficult to expect the observation of AB oscillations. On the other hand, the OMMs observed at the high magnetic fields can be treated as an ordinary fermionic subgap state featuring a bulk-like spatial distribution. As it is shown above the corresponding AB effect differs from the one for the nonlocal topological Majorana bound state, i.e. in the case of NOMMs.

To describe analytically the found features of low-energy interference transport let us proceed to the spinless system where the SC segment is represented by the Kitaev chain [3] and the arms are single-level quantum dots (QDs). The device Hamiltonian is

$$\begin{align}\hfill {\hat{H}}_{D}& =\sum\limits _{i=1,2}\,{\xi }_{i}{d}_{i}^{+}{d}_{i}-\sum\limits _{j=1}^{N-1}\left(t{a}_{j}^{+}{a}_{j+1}-{\Delta}{a}_{j}^{+}{a}_{j+1}^{+}+\mathrm{h}.\mathrm{c}.\right)\hfill \\ \hfill & \quad -\mu \sum\limits _{j=1}^{N}\left({a}_{j}^{+}{a}_{j}-\frac{1}{2}\right)-\left({\tau }_{1}{d}_{1}^{+}{a}_{1}+{\tau }_{2}{d}_{2}^{+}{a}_{N}+\mathrm{h}.\mathrm{c}.\right).\hfill \end{align} \tag{ 14 }$$

For further derivation of the conductance formula in the spinless case it is useful to separate electron and hole degrees of freedom in the normal-metal reservoir. In particular, the frequency-dependent blocks of the retarded and lesser self-energies in (7) can be represented as

$$\begin{align}\hfill {\hat{{\Sigma}}}_{ji}^{r}& =-\frac{i}{2}\mathrm{diag}\left({{\Gamma}}_{eji},{{\Gamma}}_{hji}\right)\equiv -\frac{i}{2}\left({\hat{{\Gamma}}}_{eji}+{\hat{{\Gamma}}}_{hji}\right),\enspace \hfill \\ \hfill {\hat{{\Sigma}}}_{ji}^{+-}& =i\cdot \mathrm{diag}\left({{\Gamma}}_{eji}{n}_{e},{{\Gamma}}_{hji}{n}_{h}\right)\equiv i\left({\hat{{\Gamma}}}_{eji}{n}_{e}+{\hat{{\Gamma}}}_{hji}{n}_{h}\right),\hfill \end{align} \tag{ 15 }$$

where Γ_ejj = Γ_j , Γ_e12 = Γ₁₂ e^iϕ , ${{\Gamma}}_{e21}={{\Gamma}}_{e12}^{\ast }$, ${{\Gamma}}_{hji}={{\Gamma}}_{eji}^{\ast }$. The last relations imply ${\hat{{\Gamma}}}_{e,h}={\hat{{\Gamma}}}_{e,h}^{+}$ that, in turn, results in

$$\begin{equation}\mathrm{Tr}\left[\hat{\sigma }\,\mathrm{R}\mathrm{e}\left\{{\hat{{\Sigma}}}^{+-}{\hat{G}}^{a}\right\}\right]=-\mathrm{Tr}\left[\hat{\sigma }{\hat{{\Sigma}}}^{+-}{\hat{G}}^{r}{\hat{{\Sigma}}}^{r}{\hat{G}}^{a}\right].\end{equation} \tag{ 16 }$$

Using the Keldysh equation and formula (16) the current can be represented by the Landauer-type formula, $I=2\frac{e}{h}\int \mathrm{d}\omega \,{T}_{\mathrm{L}\mathrm{A}\mathrm{R}}\left({n}_{h}-{n}_{e}\right)$, where the local Andreev reflection (LAR) probability is written as

$$\begin{equation}{T}_{\mathrm{L}\mathrm{A}\mathrm{R}}=\mathrm{Tr}\left[\mathrm{R}\mathrm{e}\left\{{\hat{{\Gamma}}}_{e}{\hat{G}}^{r}{\hat{{\Gamma}}}_{h}{\hat{G}}^{a}\right\}\right].\end{equation} \tag{ 17 }$$

The factor 2 in the expression for current appears since both electron and hole degrees of freedom participate at transport. Next, we appeal for the particular structure of the ${\hat{G}}^{r,a}$ blocks. After some mathematical calculations the LAR transmission becomes

$$\begin{equation}{T}_{\mathrm{L}\mathrm{A}\mathrm{R}}=\vert {\text{e}}^{\text{i}\phi }{{\Gamma}}_{1}{f}_{11}+{\text{e}}^{-\text{i}\phi }{{\Gamma}}_{2}{f}_{22}+{{\Gamma}}_{12}\left({f}_{12}+{f}_{21}\right){\vert }^{2},\end{equation} \tag{ 18 }$$

where ${f}_{ji}\equiv \left\langle \left\langle {d}_{j}^{+}\vert {d}_{i}^{+}\right\rangle \right\rangle$ are the Fourier transforms of anomalous retarded GFs. For the sake of brevity, hereinafter the index 'r' denoting retarded character of the response is omitted. In the following we are concentrated on a study of quantum transport in the linear-response regime at the low-temperature limit. Then, the conductance is

$$\begin{equation}G=\mathrm{d}I/\mathrm{d}V=2{G}_{0}{T}_{\mathrm{L}\mathrm{A}\mathrm{R}}.\end{equation} \tag{ 19 }$$

Thus, the analysis of transport features reduces to the finding of f_ji . In order to solve this problem the EOM technique is utilized for the three cases [46]. The corresponding details are in appendix A.

As a starting point we consider the Kitaev chain at the symmetric point, t = Δ and μ = 0, with the adjacent QDs (the symmetric Kitaev chain, SKC). In this case there are two NOMMs. The substitution of found GFs (A.5) into (18) gives

$$\begin{align}\hfill {T}_{\mathrm{N}\mathrm{O}\mathrm{M}\mathrm{M}}& =4{t}^{4}{\omega }^{2}\left[{{\Gamma}}_{1}^{2}{\tau }_{1}^{4}{D}_{2}^{2}+{{\Gamma}}_{2}^{2}{\tau }_{2}^{4}{D}_{1}^{2}\right.\hfill \\ \hfill & \quad \left.-2{{\Gamma}}_{1}{{\Gamma}}_{2}{\tau }_{1}^{2}{\tau }_{2}^{2}{D}_{1}{D}_{2}\,\mathrm{cos}\,2\phi \right]/\vert Z{\vert }^{2},\hfill \end{align} \tag{ 20 }$$

where ${D}_{j}=C\left({\omega }^{2}-{\varepsilon }_{j}^{2}-{\tau }_{j}^{2}\right)-{\tau }_{j}^{2}\left({\omega }^{2}-{\tau }_{j}^{2}\right)$, C = ω² − 4t², j = 1, 2.

It is essential to notice that if $\phi =\pi \left(n+1/2\right)$, $n\in \mathbb{Z}$, the GFs have a first-order pole at ω = 0, i.e. $Z=\omega \tilde{Z}$. Then,

$$\begin{equation}{\tilde{T}}_{\mathrm{N}\mathrm{O}\mathrm{M}\mathrm{M}}=4{t}^{4}{\left[{{\Gamma}}_{1}{\tau }_{1}^{2}{D}_{2}\,+\,{{\Gamma}}_{2}{\tau }_{2}^{2}{D}_{1}\right]}^{2}/\vert \tilde{Z}{\vert }^{2},\end{equation} \tag{ 21 }$$

where

$$\begin{align}\hfill \tilde{Z}& =\omega \left\{{D}_{1}{D}_{2}-\frac{C}{4}\left({{\Gamma}}_{1}^{2}{D}_{2}\,+\,{{\Gamma}}_{2}^{2}{D}_{1}+2{{\Gamma}}_{1}{{\Gamma}}_{2}{D}_{3}\right)\right\}\hfill \\ \hfill & \quad +i\left\{2{t}^{2}\left({{\Gamma}}_{1}{\tau }_{1}^{2}{D}_{2}\,+\,{{\Gamma}}_{2}{\tau }_{2}^{2}{D}_{1}\right)\right.\hfill \\ \hfill & \enspace \left.+{\omega }^{2}\left[{{\Gamma}}_{1}\left(C-{\tau }_{1}^{2}\right){D}_{2}\,+\,{{\Gamma}}_{2}\left(C-{\tau }_{2}^{2}\right){D}_{1}\right]\right\},\hfill \end{align} \tag{ 22 }$$

where ${D}_{3}=C\left({\omega }^{2}-{\varepsilon }_{1}{\varepsilon }_{2}-{\tau }_{1}^{2}-{\tau }_{2}^{2}\right)+{\tau }_{1}^{2}{\tau }_{2}^{2}-2{t}^{2}\left({\tau }_{1}^{2}+{\tau }_{2}^{2}\right)$. In other words, ω = 0 is an energy of the bound state in continuum. Taking into account the results (20) and (21) the conductance in the NOMM case is

$$\begin{equation}{G}_{\mathrm{N}\mathrm{O}\mathrm{M}\mathrm{M}}=\left\{\begin{array}{cc}\hfill 2{G}_{0},\hfill & \hfill \phi =\pi \left(n+1/2\right);\hfill \\ \hfill 0,\hfill & \hfill \phi \ne \pi \left(n+1/2\right).\hfill \end{array}\right.\end{equation} \tag{ 23 }$$

The revealed behavior is universal as it is independent on the ratio between ɛ_j , t_j , τ_j (j = 1, 2). The obtained results are confirmed by numerics. In figure 5(a) the AB oscillations for the SKC case are displayed by the solid curve.

Zoom In Zoom Out Reset image size

The found features of conductance coincide with the ones for the situation when there is no QDs between the normal contact and Kitaev chain. The corresponding anomalous GF's are provided in appendix A2. Using them the LAR-mediated transmission can be expressed as

$$\begin{equation}{T}_{0}=\frac{{t}^{4}{\omega }^{2}{\left({\omega }^{2}-{t}^{2}\right)}^{2}}{4\vert Z{\vert }^{2}}\left[{\Delta}{{\Gamma}}^{2}\,{\mathrm{cos}}^{2}\,\phi +{{\Gamma}}^{2}\,{\mathrm{sin}}^{2}\,\phi \right],\end{equation} \tag{ 24 }$$

where ΔΓ = Γ₁ − Γ₂, Γ = Γ₁ + Γ₂. Taking into account the phase dependence of the denominator, Z, presented in the formula (A.10) the linear-response conductance is described by the expression (23) and is not affected by the Γ₁/Γ₂ ratio.

It is worth to note that the observed sharp AB oscillations resemble the corresponding conductance behavior of two noninteracting QDs (double quantum dot, DQD) between normal contacts in the limiting case ɛ₁ = ɛ₂ = 0 (see black dotted curve in figure 5(a)) [47]. Thus, in the separate SKC (i.e. without the side-attached dots) two decoupled Majorana fermions, γ_1A and γ_NB , can be treated as such a double-dot structure connected to the electron and hole reservoirs in parallel. However, it is essential that in the case of Majoranas the G maxima are fundamentally shifted by the π/2 in comparison with the mentioned DQD situation [47] (see blue dashed and black dotted curves in figure 5(a)). This effect is explained by the fact that the same phase carriers acquire tunneling from the contact to the B-type Majorana mode in the bottom arm. Note that the similar effect was demonstrated in the nonlocal transport through the QD with the side-coupled SKC having the Π-shape [32, 33].

In general, the suggested analogy with two noninteracting QDs is not complete since in our case the other eigenstates having nonzero energies are also able to contribute to the transport. For example, in the simplest situation of SKC directly coupled with the contact two Majorana double-dot molecules (consisting of γ_1B , γ_2A and γ_N−1,B , γ_NA , respectively) also interact with it. Then, both arms of the device include two chains characterized by opposite parity of Majorana-fermion number. Taking it into account in appendix B we numerically analyzed the transport properties of the analogous AB interferometer containing ordinary zero-energy QDs. It is shown the result (23) can be reproduced for any odd number of QDs in the separate arm (see appendix B for details).

Hence, it becomes clear why there is no effect of nonzero ɛ₁, ɛ₂ on the conductance features, that can be perceived as another manifestation of the Majorana-mode leakage [43]. The explicit reason of such a behavior is uncovered if the QD–SKC–QD device is considered in the Majorana-operator representation. As it is shown in the top row of figure 5(b) for any ɛ_1,2 there are two uncoupled chains characterized by an odd-parity number of the Majorana fermions connected in series. That, in turn, implies the presence of the two zero-energy eigenstates interacting with the normal contact. Obviously, the oscillations (23) have to keep for the arbitrary number of zero-energy QDs attached to the SKC edges.

Now we focus on the SC wire in the nontrivial phase modeled by the effective Hamiltonian ${\hat{H}}_{\mathrm{e}\mathrm{ff}}=i\xi {b}_{1}{b}_{2}/2$, where ξ ∼ e^−N is a measure of hybridization of the Majorana modes ${b}_{1}=\sum _{j=1}^{N}{w}_{j}{\gamma }_{jA}$ and ${b}_{2}=\sum _{j=1}^{N}{z}_{j}{\gamma }_{jB}$ (b₁ = γ_1A , b₂ = γ_NB if N → ∞) [3]. As this interaction is supposed to be weak there is no immediate coupling between b₁ (b₂) and the right (left) QD (see the middle row of figure 5(b)). Then, based on the necessary GFs which are brought in appendix A3, the conductance in the symmetric transport regime (i.e. Γ₁ = Γ₂ = Γ/2, τ₁ = τ₂ = τ, ɛ₁ = ɛ₂ = ɛ) has form

$$\begin{equation}{G}_{\mathrm{e}\mathrm{ff}}=2{G}_{0}\cdot {{\Gamma}}^{2}{\tau }^{4}{\omega }^{2}\frac{{\left({\omega }^{2}-{\varepsilon }^{2}-{\tau }^{2}\right)}^{2}{\mathrm{sin}}^{2}\,\phi }{4\vert Z{\vert }^{2}}.\end{equation} \tag{ 25 }$$

From expression (A.14) one can easily conclude that for the NOMMs (ξ = 0) the conductance at ω = 0 is equal to 2G₀ if $\phi =\pi \left(n+1/2\right)$, $n\in \mathbb{Z}$. Otherwise, it is zero, that is in complete agreement with the above-obtained result (23). However, the mentioned features also take place if ɛ = 0 even though ξ ≠ 0. The latter can be easily explained if one again refers to the system in the Majorana representation displayed in the middle row of figure 5(b). Indeed, when on-dot energies ɛ_1,2 are zero two unpaired Majorana fermions, γ_1A and γ_2B , occur giving rise to one zero-energy state in each arm. In general situation of ɛ, ξ ≠ 0 the linear-response conductance is zero. Mathematically, it means that the first-order pole of the GFs arises at ω = 0 only if $\mathrm{cos}\,\phi =2\varepsilon \xi \left(\pm 2i\varepsilon /{\Gamma}-1\right)/{\tau }^{2}$.

Finally, far from the symmetric point of the Kitaev model, the Majorana zero modes become to hybridize significantly. In particular, it infers that the mode, b₁ or b₂, initially localized at one of the chain's edges shows the nonzero probability density at the opposite edge. It results in the direct coupling to both QDs with the different amplitudes proportional to ${\tau }_{1,2}\left(u\pm v\right)$ or ${\tau }_{1,2}\left(u\mp v\right)$ (see the dashed and dash-dotted curves in the bottom row of figure 5(b), respectively) [48, 49]. If t/Δ ≫ 1 there are N values of the chemical potential (A.15) at which the zero-energy Bogoliubov excitation with the OMMs emerges. Solving the EOMs (A.20) presented in appendix A4 one can obtain the following transmission in the symmetric transport regime:

$$\begin{equation}{T}_{\mathrm{O}\mathrm{M}\mathrm{M}}=\frac{4{{\Gamma}}^{2}{\tau }_{e}^{2}{\tau }_{h}^{2}{\omega }^{2}{\left({\omega }^{2}-{\varepsilon }^{2}-2{\tau }_{p}\right)}^{2}{\mathrm{sin}}^{2}\,\phi }{\vert Z{\vert }^{2}}.\end{equation} \tag{ 26 }$$

Taking into account the ω-dependence of the GFs' denominator (A.21) the linear-response conductance is

$$\begin{equation}{G}_{\mathrm{O}\mathrm{M}\mathrm{M}}=\left\{\begin{array}{cc}\hfill 2{G}_{0},\hfill & \hfill \,\,\,\,\,\varepsilon =0\enspace \text{and}\enspace v/u=\mathrm{tan}\,\phi /2;\hfill \\ \hfill 0,\hfill & \hfill \,\,\,\,\,\varepsilon \ne 0\enspace \text{or}\enspace v/u\ne \mathrm{tan}\,\phi /2.\hfill \end{array}\right.\end{equation} \tag{ 27 }$$

The AB oscillations are depicted in figure 5(a) by the dashed curve. According to the expression (27) the maxima appear at the phase values ϕ = 2 arctan v/u (see the graphic solution of the equation v/u = tan ϕ/2 in the inset). Note that the positions of conductance peaks tend to $\phi =\pi \left(n+1/2\right)$ if the hybridization of the Majorana modes vanishes meaning the condition u = v. Hence, in the OMM situation the AB oscillation period doubles (compare solid and dashed curves in figure 5(a)).

The difference between ɛ ≠ 0 and ɛ = 0 cases can be readily understood looking at the bottom row of figure 5(b). If ɛ = 0 two noninteracting chains emerge each of which is a triple Majorana-dot molecule coupled in parallel with the electron and hole reservoirs. In other words, it again provides the doubly degenerate zero-energy eigenstate of the device. Such a state is absent when ɛ ≠ 0.

In order to treat the transport problem analytically we first address to the Kitaev model at the symmetric point, t = Δ and μ = 0. Employing the Majorana-operator representation, ${a}_{j}=\frac{1}{2}\left({\gamma }_{jA}+i{\gamma }_{jB}\right)$, where ${\gamma }_{jA,B}={\gamma }_{jA,B}^{+}$, one can rewrite the device Hamiltonian (14) in the form

$$\begin{align}\hfill {\hat{H}}_{D}& =\sum\limits _{i=1,2}\,{\xi }_{i}{d}_{i}^{+}{d}_{i}+it\sum\limits _{j=1}^{N-1}\,{\gamma }_{jB}{\gamma }_{j+1,A}\hfill \\ \hfill & \quad -\frac{{\tau }_{1}}{2}\left[\left({d}_{1}^{+}-{d}_{1}\right){\gamma }_{1A}+i\left({d}_{1}^{+}+{d}_{1}\right){\gamma }_{1B}\right]\hfill \\ \hfill & \quad -\frac{{\tau }_{2}}{2}\left[\left({d}_{2}^{+}-{d}_{2}\right){\gamma }_{NA}+i\left({d}_{2}^{+}+{d}_{2}\right){\gamma }_{NB}\right].\hfill \end{align} \tag{ A.3 }$$

As a result, the EOM system (A.1) for i = 1 is ${\hat{A}}_{1}{\hat{x}}_{1}={\hat{B}}_{1}$, where ${\hat{x}}_{1}={\left({g}_{11},{g}_{21},{f}_{11},{f}_{21}\right)}^{\mathrm{T}}$. The coefficient matrix and column vector are

$$\begin{align}\hfill {\hat{A}}_{1}& =\left(\begin{matrix}\hfill {z}_{1e}\hfill & \hfill {z}_{12}\hfill & \hfill 2{t}^{2}{\tau }_{1}^{2}\hfill & \hfill 0\hfill \\ \hfill -{z}_{12}^{\ast }\hfill & \hfill {z}_{2e}\hfill & \hfill 0\hfill & \hfill -2{t}^{2}{\tau }_{2}^{2}\hfill \\ \hfill 2{t}^{2}{\tau }_{1}^{2}\hfill & \hfill 0\hfill & \hfill {z}_{1h}\hfill & \hfill -{z}_{12}^{\ast }\hfill \\ \hfill 0\hfill & \hfill -2{t}^{2}{\tau }_{2}^{2}\hfill & \hfill {z}_{12}\hfill & \hfill {z}_{2h}\hfill \end{matrix}\right),\hfill \\ \hfill {\hat{B}}_{1}& ={\left(\omega C,0,0,0\right)}^{\mathrm{T}},\hfill \end{align} \tag{ A.4 }$$

where ${z}_{je\left(h\right)}=C\left(\omega {C}_{je\left(h\right)}-{\tau }_{j}^{2}\right)-2{t}^{2}{\tau }_{j}^{2}$, z₁₂ = iωCΓ₁₂ e^iϕ /2, C = ω² − 4t², ${C}_{je\left(h\right)}=\omega \mp {\varepsilon }_{j}+i{{\Gamma}}_{j}/2$.

The EOM system for i = 2 can be written in a similar manner. Their solution gives the sought anomalous GFs,

$$\begin{align}\hfill & {f}_{11}=-\frac{{t}^{2}}{2}\cdot \frac{\omega C{\tau }_{2}^{2}{{\Gamma}}_{12}^{2}\,{\mathrm{e}}^{-2\mathrm{i}\phi }+4{\tau }_{1}^{2}{Z}_{2}}{Z},\hfill \\ \hfill & {f}_{22}=\frac{{t}^{2}}{2}\cdot \frac{\omega C{\tau }_{1}^{2}{{\Gamma}}_{12}^{2}\,{\mathrm{e}}^{2\mathrm{i}\phi }+4{\tau }_{2}^{2}{Z}_{1}}{Z},\hfill \\ \hfill & {f}_{21}=i{{\Gamma}}_{12}{t}^{2}\cdot \frac{{\tau }_{1}^{2}{z}_{2e}\,{\text{e}}^{\text{i}\phi }-{\tau }_{2}^{2}{z}_{1h}\,{\text{e}}^{-\text{i}\phi }}{Z},\hfill \\ \hfill & {f}_{12}=i{{\Gamma}}_{12}{t}^{2}\cdot \frac{{\tau }_{1}^{2}{z}_{2h}\,{\text{e}}^{\text{i}\phi }-{\tau }_{2}^{2}{z}_{1e}\,{\text{e}}^{-\text{i}\phi }}{Z},\hfill \end{align} \tag{ A.5 }$$

where

$$\begin{align}\hfill {Z}_{j}& =\omega \left(C{C}_{je}{C}_{jh}+{\tau }_{j}^{4}\right)-{\tau }_{j}^{2}(C+2{t}^{2})({C}_{je}+{C}_{jh}),\hfill \\ \hfill Z& ={Z}_{1}{Z}_{2}+\frac{{{\Gamma}}_{12}^{2}}{4}\left({z}_{1e}{z}_{2e}+{z}_{1h}{z}_{2h}+{\omega }^{2}{C}^{2}\frac{{{\Gamma}}_{12}^{2}}{4}\right.\hfill \\ \hfill & \quad \left.-8{t}^{4}{\tau }_{1}^{2}{\tau }_{2}^{2}\,\mathrm{cos}\,2\phi \right).\hfill \end{align} \tag{ A.6 }$$

If there are no dots between the contact and Kitaev chain, then in the symmetric point one can easily write the following coefficient matrix and column vector writing the EOMs for the Majorana GFs, ${\hat{A}}_{1A}{\hat{x}}_{1A}={\hat{B}}_{1A}$, where ${\hat{x}}_{1A}={\left({m}_{1A,1A},{m}_{1B,1A},{m}_{NA,1A},{m}_{NB,1A}\right)}^{\mathrm{T}}$ and ${m}_{iA,jB}=\left\langle \left\langle {\gamma }_{iA}\vert {\gamma }_{jB}\right\rangle \right\rangle$:

$$\begin{align}\hfill & {\hat{A}}_{1A}=\left(\begin{matrix}\hfill {\hat{D}}_{1}\hfill & \hfill {\hat{T}}_{12}\hfill \\ \hfill {\hat{T}}_{12}^{T}\hfill & \hfill {\hat{D}}_{2}\hfill \end{matrix}\right),\hspace{25.0pt}{\hat{T}}_{12}=2i{{\Gamma}}_{12}\omega \left(\begin{matrix}\hfill \mathrm{cos}\,\phi \hfill & \hfill -\mathrm{sin}\,\phi \hfill \\ \hfill \mathrm{sin}\,\phi \hfill & \hfill \mathrm{cos}\,\phi \hfill \end{matrix}\right),\hfill \\ \hfill & {\hat{D}}_{1}=\mathrm{diag}\left({{\Omega}}_{1},{\omega }_{1}\right),\hspace{25.0pt}{\hat{D}}_{2}=\mathrm{diag}\left({\omega }_{2},{{\Omega}}_{2}\right),\hfill \\ \hfill & {\hat{B}}_{1A}={\left(2\omega ,0,0,0\right)}^{\mathrm{T}},\hfill \end{align} \tag{ A.7 }$$

where Ω_j = ωω_j − t², ω_j = ω + 2iΓ_j . Note that the factor 2 in ${\hat{B}}_{1A}$ is caused by the nonfermionic anticommutation relations, i.e. $\left\{{\gamma }_{iA\left(B\right)},{\gamma }_{jA\left(B\right)}\right\}=2{\delta }_{ij}$. The three systems for the rest of Majorana GFs, i.e. ${m}_{jA\left(B\right),1B}$, ${m}_{jA\left(B\right),2A}$ and ${m}_{jA\left(B\right),2B}$, have similar form. Finally, taking into account the relation between the Majorana and fermionic GF's,

$$\begin{equation}{f}_{ij}=\frac{1}{4}\left[{m}_{iA,jA}-{m}_{iB,jB}-i\left({m}_{iA,jB}+{m}_{iB,jA}\right)\right],\end{equation} \tag{ A.8 }$$

the required anomalous GFs are

$$\begin{align}\hfill & {f}_{11}=-\frac{{t}^{2}}{2}\cdot \frac{{{\Omega}}_{2}\left(\omega +2i{{\Gamma}}_{2}\right)+4{{\Gamma}}_{12}^{2}\omega \,{\mathrm{e}}^{2\mathrm{i}\phi }}{Z},\hfill \\ \hfill & {f}_{22}=\frac{{t}^{2}}{2}\cdot \frac{{{\Omega}}_{1}\left(\omega +2i{{\Gamma}}_{1}\right)+4{{\Gamma}}_{12}^{2}\omega \,{\mathrm{e}}^{-2\mathrm{i}\phi }}{Z},\hfill \end{align} \tag{ A.9 }$$

$$\begin{align}\hfill & {f}_{12}={f}_{21}={{\Gamma}}_{12}{t}^{2}\cdot \frac{2{\Delta}{\Gamma}\omega \,\mathrm{cos}\,\phi +\left[\omega \left({\omega }_{1}+{\omega }_{2}\right)-{t}^{2}\right]\mathrm{sin}\,\phi }{Z},\hfill \\ \hfill & Z=\omega {\Omega}\left[\omega {\Omega}-2i{\Gamma}{t}^{2}\right]-4{{\Gamma}}_{12}^{2}{t}^{4}\,{\mathrm{cos}}^{2}\,\phi ,\hfill \end{align} \tag{ A.10 }$$

where Γ = Γ₁ + Γ₂, ΔΓ = Γ₁ − Γ₂, Ω = ω² + 2iΓω − t².

Let us turn to the effective model of the Kitaev chain in the nontrivial phase which describes two Majorana modes localized at the opposite ends on the chain and weakly interacting with each other [3]. In this case the device Hamiltonian can be written as

$$\begin{align}\hfill {\hat{H}}_{D}& =\sum\limits _{i=1,2}\,{\xi }_{i}{d}_{i}^{+}{d}_{i}+i\frac{\xi }{2}{b}_{1}{b}_{2}\hfill \\ \hfill & \quad -\frac{{\tau }_{1}}{2}\left({d}_{1}^{+}-{d}_{1}\right){b}_{1}-i\frac{{\tau }_{2}}{2}\left({d}_{2}^{+}+{d}_{2}\right){b}_{2},\hfill \end{align} \tag{ A.11 }$$

where ξ—a hybridization parameter. Hence, the EOM technique leads to

$$\begin{align}\hfill & {\hat{A}}_{1}=\left(\begin{matrix}\hfill {z}_{1e}\hfill & \hfill {Z}_{12}\hfill & \hfill {T}_{1}\hfill & \hfill {T}_{12}\hfill \\ \hfill -{Z}_{12}^{\ast }\hfill & \hfill {z}_{2e}\hfill & \hfill -{T}_{12}\hfill & \hfill {T}_{2}\hfill \\ \hfill {T}_{1}\hfill & \hfill -{T}_{12}\hfill & \hfill {z}_{1h}\hfill & \hfill -{Z}_{12}^{\ast }-2{T}_{12}\hfill \\ \hfill {T}_{12}\hfill & \hfill {T}_{2}\hfill & \hfill {Z}_{12}-2{T}_{12}\hfill & \hfill {z}_{2h}\hfill \end{matrix}\right)\hfill \\ \hfill & {\hat{B}}_{1}={\left({\omega }^{2}-{\xi }^{2},0,0,0\right)}^{\mathrm{T}},\hfill \end{align} \tag{ A.12 }$$

where ${z}_{1e\left(h\right)}=\left({\omega }^{2}-{\xi }^{2}\right){C}_{1e\left(h\right)}-{T}_{1}$, ${z}_{2e\left(h\right)}=\left({\omega }^{2}-{\xi }^{2}\right){C}_{2e\left(h\right)}-{T}_{2}$, T₁₂ = −ξτ₁ τ₂/2, ${T}_{j}=\omega {\tau }_{j}^{2}/2$, ${Z}_{12}=i\left({\omega }^{2}-{\xi }^{2}\right){{\Gamma}}_{12}\,{\text{e}}^{\text{i}\phi }/2+{T}_{12}$.

Since in the general case ${\hat{A}}_{1}$ has no zero elements the solution of EOM system ${\hat{A}}_{1}{\hat{x}}_{1}={\hat{B}}_{1}$ is more cumbersome. We adduce it for the symmetric transport configuration when Γ₁ = Γ₂ = Γ/2, τ₁ = τ₂ = τ, ɛ₁ = ɛ₂ = ɛ,

$$\begin{align}\hfill {f}_{11\left(22\right)}& =\pm \frac{{\tau }^{2}}{4Z}\left[4\omega \left({\omega }^{2}-{\tau }^{2}-{\varepsilon }^{2}\right)\right.\hfill \\ \hfill & \quad \left.+i{\Gamma}\left(2{\omega }^{2}-{\tau }^{2}-2\varepsilon \xi \,{\mathrm{e}}^{\mp \mathrm{i}\phi }\mp \frac{{\Gamma}}{2}\omega \,{\mathrm{e}}^{\mp \mathrm{i}\phi }\,\mathrm{sin}\,\phi \right)\right],\hfill \\ \hfill {f}_{21\left(12\right)}& =\mp \frac{{\tau }^{2}}{4Z}\left[4\xi \left({\omega }^{2}-{\varepsilon }^{2}+i{\Gamma}\omega \right)-2i{\Gamma}\varepsilon \omega \,\mathrm{cos}\,\phi \right.\hfill \\ \hfill & \quad \left.\mp {\Gamma}\left(\omega \left(2\omega +i\frac{{\Gamma}}{2}\right)-{\tau }^{2}\right)\,\mathrm{sin}\,\phi \right],\hfill \end{align} \tag{ A.13 }$$

where

$$\begin{align}\hfill Z& =\frac{{{\Gamma}}^{2}}{8}{\tau }^{4}\,{\mathrm{cos}}^{2}\,\phi -{\tau }^{4}\omega \left(2\omega +i{\Gamma}\right)\hfill \\ \hfill & \quad -2\left({\omega }^{2}-{\xi }^{2}\right)\left({\omega }^{2}-{\varepsilon }^{2}\right)\left[{\left(\omega +i\frac{{\Gamma}}{2}\right)}^{2}-{\varepsilon }^{2}\right]\hfill \\ \hfill & \quad +{\tau }^{2}\left(4\omega +i{\Gamma}\right)\left[\omega \left({\omega }^{2}+i\frac{{\Gamma}}{2}\omega -{\varepsilon }^{2}\right)-i\frac{{\Gamma}}{2}\varepsilon \xi \,\mathrm{cos}\,\phi \right].\hfill \end{align} \tag{ A.14 }$$

Next, we analyze transport in the device where two QDs coupled in parallel with the single zero-energy Bogoliubov excitation characterized by the significant overlap of the Majorana wave functions. It is well known that the Kitaev chain with N sites has the same number of lines in the parametric space where the lowest excitation energy becomes zero [50], which are defined by the following equations:

$$\begin{equation}\mu =2\sqrt{{t}^{2}-{{\Delta}}^{2}}\,\mathrm{cos}\,\frac{\pi m}{N+1},\quad m=1,\dots ,N.\end{equation} \tag{ A.15 }$$

The Bogoliubov operators of these zero-energy states, α_m , have form [51]

$$\begin{align}\hfill {\alpha }_{m}& =\frac{1}{{S}_{m}}\sum\limits _{j=1}^{N}{r}^{j-1}\,\mathrm{sin}\,\frac{\pi mj}{N+1}\hfill \\ \hfill & \quad \cdot \left({a}_{j}+{a}_{j}^{+}+{a}_{N+1-j}-{a}_{N+1-j}^{+}\right),\hfill \end{align} \tag{ A.16 }$$

where S_m —a normalization factor; $r=\sqrt{\frac{t-{\Delta}}{t+{\Delta}}}$. For concreteness, we consider one of such states, α_m=1 ≡ α. As a result, the u_j and v_j (j = 1, ..., N/2) coefficients are

$$\begin{align}\hfill & {u}_{j}={u}_{N+1-j}=\frac{1}{S}{r}^{j-1}\left(1+{r}^{N-j}\right)\,\mathrm{sin}\,\frac{\pi j}{N+1},\hfill \\ \hfill & {v}_{j}=-{v}_{N+1-j}=\frac{1}{S}{r}^{j-1}\left(1-{r}^{N-j}\right)\,\mathrm{sin}\,\frac{\pi j}{N+1}.\hfill \end{align} \tag{ A.17 }$$

Note if μ = 0, then the equations (A.15) and (A.17) obviously lead to the SKC and u = v. Consequently, the Majorana-mode operators are ${b}_{1}={\alpha }^{+}+\alpha ={\sum }_{j}{w}_{j}{\gamma }_{jA}=2u\left({a}_{1}^{+}+{a}_{1}\right)$, ${b}_{2}=i\left({\alpha }^{+}-\alpha \right)={\sum }_{j}{z}_{j}{\gamma }_{jB}=2iu\left({a}_{N}^{+}-{a}_{N}\right)$. In opposite, if μ ≠ 0, then the wave functions related to b₁ and b₂ overlap. The hybridization is stronger if t/Δ ≫ 1 resulting in the zero-energy excitation with the bulk spatial distribution.

The inverse Bogoliubov transform gives [30]

$$\begin{equation}{a}_{1}\approx u\alpha +v{\alpha }^{+},\hspace{25.0pt}{a}_{N}\approx u\alpha -v{\alpha }^{+},\end{equation} \tag{ A.18 }$$

where u ≡ u₁, v ≡ v₁. Then, the structure Hamiltonian is

$$\begin{align}\hfill {\hat{H}}_{D}& =\sum\limits _{i=1,2}{\xi }_{i}{d}_{i}^{+}{d}_{i}-\left({\tau }_{1e}{d}_{1}^{+}+{\tau }_{2e}{d}_{2}^{+}\right)\alpha \hfill \\ \hfill & \quad -\left({\tau }_{1h}{d}_{1}^{+}-{\tau }_{2h}{d}_{2}^{+}\right){\alpha }^{+}+\mathrm{h}.\mathrm{c}.,\hfill \end{align} \tag{ A.19 }$$

where τ_je = τ_j u, τ_jh = τ_j v.

The EOM coefficient matrix and column vector can be written as

$$\begin{align}\hfill {\hat{A}}_{1}& =\left(\begin{matrix}\hfill {z}_{1e}\hfill & \hfill {Z}_{12}\hfill & \hfill 2{T}_{1}\hfill & \hfill {T}_{eh}\hfill \\ \hfill -{Z}_{12}^{\ast }-2{T}_{12}\hfill & \hfill {z}_{2e}\hfill & \hfill {T}_{eh}\hfill & \hfill -2{T}_{2}\hfill \\ \hfill 2{T}_{1}\hfill & \hfill {T}_{eh}\hfill & \hfill {z}_{1h}\hfill & \hfill -{Z}_{12}^{\ast }-2{T}_{12}\hfill \\ \hfill {T}_{eh}\hfill & \hfill -2{T}_{2}\hfill & \hfill {Z}_{12}\hfill & \hfill {z}_{2h}\hfill \end{matrix}\right),\hfill \\ \hfill {\hat{B}}_{1}& ={\left(\omega ,0,0,0\right)}^{\mathrm{T}},\hfill \end{align} \tag{ A.20 }$$

where ${z}_{je\left(h\right)}=\omega {C}_{je\left(h\right)}-{\tau }_{je}^{2}-{\tau }_{jh}^{2}$, T_j = τ_je τ_jh , T_eh = τ_1h τ_2e − τ_1e τ_2h , T₁₂ = τ_1e τ_2e − τ_1h τ_2h , Z₁₂ = iΓ₁₂ e^iϕ /2 − T₁₂. The EOM system solution in the symmetric transport regime yields

$$\begin{align*}\hfill {f}_{11\left(22\right)}& =\mp \frac{2{\tau }_{e}{\tau }_{h}}{Z}\left[i{\Gamma}\,{\text{e}}^{-\text{i}\phi }\left(2{\tau }_{m}\mp \frac{{\Gamma}}{2}\omega \,\mathrm{sin}\,\phi \right)\right.\hfill \\ \hfill & \quad \left.+2\omega \left(\omega C-{\varepsilon }^{2}\right)-2{\tau }_{p}\left(\omega +C\right)\right],\hfill \\ \hfill {f}_{21\left(12\right)}& =\frac{2{\tau }_{e}{\tau }_{h}}{Z}\left[{\Gamma}\left(\left({\tau }_{p}-\frac{\omega }{2}\left(\omega +C\right)\right)\mathrm{sin}\,\phi \right.\right.\hfill \\ \hfill & \quad \left.\left.-i\varepsilon \omega \,\mathrm{cos}\,\phi \right)\pm 4{\tau }_{m}\varepsilon \right],\hfill \end{align*}$$

where C = ω + iΓ/2, ${\tau }_{p\left(m\right)}={\tau }_{e}^{2}\pm {\tau }_{h}^{2}$. The GFs' denominator is

$$\begin{align}\hfill Z& ={\omega }^{2}\left({\omega }^{2}-{\varepsilon }^{2}\right)\left({C}^{2}-{\varepsilon }^{2}\right)\hfill \\ \hfill & \quad +2\omega \left[i\frac{{\Gamma}}{2}{\tau }_{m}\left(\omega C+{\varepsilon }^{2}\right)\mathrm{cos}\,\phi -{\tau }_{p}\left(\omega +C\right)\left(\omega C-{\varepsilon }^{2}\right)\right]\hfill \\ \hfill & \quad +{\left[{\tau }_{p}\left(\omega +C\right)\mathrm{cos}\,\phi -i\frac{{\Gamma}}{2}{\tau }_{m}\right]}^{2}+4{\tau }_{p}^{2}\omega C\,{\mathrm{sin}}^{2}\,\phi -4{\tau }_{m}^{2}{\varepsilon }^{2}.\hfill \end{align} \tag{ A.21 }$$