Majorana zero modes induced by superconducting phase bias

Driving a constant current through at SC causes its phase to increase linearly [30]. This was exploited by Romito et al [48], who studied the effect of supercurrent on the Majorana nanowires of [33, 34]. For context, we first review this model in the absence of current.

Consider a semiconducting spin-orbit-coupled wire, which is placed in proximity to an s-wave SC, with a magnetic field applied along the wire. The BdG Hamiltonian describing the system is [33, 34]

$\begin{equation} {\cal H}_\textrm{NW} = \left( \frac{p^2}{2m}+up\sigma_x-\mu \right)\tau_z + E_\textrm{Z}\sigma_z + \Delta\tau_x, \end{equation} \tag{ 3 }$

where the Nambu spinor is the same as in equation (2), and the Pauli matrices $\sigma,\tau$ act in spin and particle-hole spaces, respectively. m is the effective mass, u is the SOC constant, µ is the chemical potential, $E_\textrm{Z}$ is the Zeeman energy induced by the magnetic field, and Δ is the pair potential, which is chosen without loss of generality to be purely real. The normal-state spectrum ($\Delta = 0$) consists of two parabolas that are shifted in momentum space according to the spin ($|{\rightarrow}\rangle$ or $|{\leftarrow}\rangle$ in the basis of σ_x eigenstates). The Zeeman term $E_\textrm{Z}\sigma_z$ does not commute with σ_x and therefore opens a gap in the intersection points of the parabolas. Therefore, the position of the chemical potential µ determines whether there are two or four Fermi points. In the case of two Fermi points, introducing $\Delta\neq0$ opens an inverted superconducting gap, making the system a topological superconductor. In terms of the model parameters, the topological region appears at $E_\textrm{Z}\gt\Delta$, i.e. the Zeeman energy must exceed the pair potential [33, 34].

Let us now add current into the mix. The only change in the Hamiltonian compared to equation (3) comes from the spatial variation of the SC phase $\phi(x)$, which changes the pairing terms to [48]

$\begin{align} {\cal H}_\textrm{SC} = \Delta\left[\tau_x\cos\phi(x) +\tau_y\sin\phi(x) \right]. \end{align} \tag{ 4 }$

To make progress, it is very useful to introduce the gauge transformation

$\begin{align} {\cal U} = \exp\left[i\frac{\phi(x)}{2}\tau_z\right]. \end{align} \tag{ 5 }$

This transformation eliminates the phase variation from the pairing terms, at the cost of shifting the momentum in the normal-state terms, $p\rightarrow p-\tau_z\nabla\phi/2$. This seemingly innocent gauge transformation actually plays a very important role in understanding the mechanism by which phase variation changes the picture, as it establishes the similarity between a phase gradient and a vector potential. Notice that, as in the case of a real vector potential, the momentum shift is opposite for electrons and holes.

For the case of a constant current we get a constant phase gradient, and therefore the momentum shift is position independent. Then, the Hamiltonian may still be analyzed in momentum space (translation invariance is restored), and the topological phase boundaries are found by the closing of the gap at p = 0. The topological phase diagram is shown in figure 2. Remarkably, the phase gradient gives rise to a topological phase at $E_\textrm{Z}\lt\Delta$, which would not be possible in its absence. Notice, however, that as the current increases the superconducting gap decreases, until it eventually vanishes (see also [31] for an elementary treatment of the gap's dependence on the phase gradient). This makes it impossible to tune into a topological phase at $E_\textrm{Z} = 0$.

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Rather than applying a current to continuously vary the SC phase, it is possible to phase bias several SCs such that a discrete collection of phases is formed. The properties of the Josephson junctions formed between two such SCs strongly depend on the phase bias. As we show below, this dependence can greatly assist in engineering topological SC.

The first idea for utilizing Josephson junctions as Majorana platforms is due to Fu and Kane [42], who pointed it out in the same work reviewed in section 2. There, two SCs are deposited on top of a topological insulator, forming an STIS junction. When the separation between the SCs is much smaller than $v/\Delta$ (where v is the Fermi velocity and Δ is the SC gap), the junction can be considered as quasi 1D, and the dispersion is approximated as

$\begin{equation} E(k) = \pm\sqrt{v^2 k^2 + \Delta^2~\cos^2\left(\frac{\phi}{2}\right)}, \end{equation} \tag{ 6 }$

where φ is the phase difference between the SCs. Crucially, at $\phi = \pi$ the system becomes gapless, which motivates the study of a low-energy effective theory near $\phi = \pi$, k = 0. The effective Hamiltonian reads

$\begin{equation} \tilde{\cal H} = -i\tilde{v}v_x\partial_x + \tilde{\Delta}v_y, \end{equation} \tag{ 7 }$

where the effective Fermi velocity $\tilde{v}$ depends on the chemical potential µ and the junction's width W, and the renormalized pair potential $\tilde{\Delta} = \Delta\cos(\phi/2)$. The Pauli matrices v act in the two-dimensional low-energy subspace. The effective Hamiltonian has the same form as the Su–Schrieffer–Heeger (SSH) model [49] and the Kitaev chain [23], and therefore at $\tilde{v} = \tilde{\Delta}$ it undergoes a topological phase transition, accompanied by the appearance of localized Majorana end states. However, it should be notes that the surface states at the sides of the 3DTI may mix with the MZMs, which is clearly detrimental in practice. An attempt to overcome this problem was made by Potter and Fu [50], utilizing Josephson vortices.

The STIS junction scheme is remarkably simple and elegant, but it is very challenging to realize experimentally, due to the difficulty in establishing a stable superconducting proximity effect in a 3DTI surface. To overcome this difficulty, two theoretical works [35, 36] proposed to exchange the 3DTI with a standard spin-orbit-coupled 2D electron gas (2DEG). This trading is possible since the Aharonov–Casher phase [51], arising from the Rashba spin–orbit coupling is actually the same as the π Berry phase associated with the Dirac cone in the TI.

Consider the setup depicted in figure 3(a). A Josephson junction of phase difference φ is formed on top of a spin-orbit-coupled 2DEG, and an in-plane magnetic field is applied. The Hamiltonian describing the system is

$\begin{align} {\cal H}_\textrm{JJ} & = \left[ -\frac{\partial_x^2+\partial_y^2}{2m} - \mu + i\alpha\left(\partial_x \sigma_y - \partial_y \sigma_x \right) \right]\tau_z \nonumber \\ &\quad + E_\textrm{Z}\sigma_x + \Delta(y)\left[\tau_x\cos\phi(y) + \tau_y\sin\phi(y) \right]. \end{align} \tag{ 8 }$

Here α is the Rashba SOC strength, and $E_\textrm{Z} = g\mu_\textrm{B}B_{\parallel}/2$ is the Zeeman energy, where g is the g-factor, µ_B is the Bohr magneton, and $B_{\parallel}$ is the in-plane magnetic field, which we take to be in the x direction. The SC pairing amplitude is $\Delta(y) = \Delta \Theta(|y|-W/2)$ and the phase is

$\begin{equation} \phi(y) = \left\{\begin{array}{ll} 0 & \textrm{for}\ y \lt 0\\ \phi & \textrm{for}\ y\gt0 \end{array}.\right. \end{equation} \tag{ 9 }$

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Some insight into the topological behavior of the system may be gained by analyzing the Hamiltonian (8) using the scattering formalism at $k_x = 0$ [36]. This is done by writing the expression for the Andreev reflection at the NS interface,

$\begin{equation} r_\textrm{A}(E) = \exp\left( i\cos^{-1}\frac{E-E_\textrm{Z,S}}{\Delta} + i\frac{\phi}{2} \right), \end{equation} \tag{ 10 }$

and the expression for the electrons and holes transmission in the normal region,

$\begin{equation} t_{e(h)}(E) = \exp\left( \pm ik_\textrm{F}{W} + i\frac{E-E_\textrm{Z,N}}{v_\textrm{F}}W \right). \end{equation} \tag{ 11 }$

Here $E_\textrm{Z,S},E_\textrm{Z,N}$ are the Zeeman energies in the superconducting and normal regions, respectively (these differ in general due to different g-factors in the different regions), and W is the junction's width. In the absence of normal reflection at the NS interface, the condition for a bound state at energy E is $t_e(E) t_h(E) r_\textrm{A}^2(E) = 1$. For $E_\textrm{Z,S} = 0$, a zero-energy state, signaling a topological phase transition (since we are looking at $k_x = 0$), appears for

$\begin{equation} \frac{E_\textrm{Z,N}}{E_\textrm{T}} \pm \frac{\phi}{\pi} = 1 + 2n, \end{equation} \tag{ 12 }$

where $E_\textrm{T} = \pi v_\textrm{F}/2W$ is the Thouless energy. This diamond-like phase diagram is shown in figure 3(b). Including also normal reflection and $E_\textrm{Z,S}\neq0$ smears the sharp edges of the diamonds. Notice that even though the SOC does not appear in equation (12), its presence is essential to the existence of the topological phase: without SOC, there would be no gap in the topological phase.

One of the important advantages of this scheme is its weak sensitivity to the chemical potential. Moreover, misalignment of the magnetic field in the plane has no dramatic effects, making it in principle more robust than nanowires. Importantly, the energy scale to which the Zeeman energy should be compared is the Thouless energy $E_\textrm{T}\propto1/W$. This sets the scale for a topological transition near φ = 0, but it is also relevant at larger value of φ. The fact that $E_\textrm{T}$ is inversely proportional to W motivates the use of wide junctions, since those would require small magnetic fields to become topological. However, narrow junctions are generally preferred as they allow quasi-ballistic motion. These two competing effects are the experimental challenges in realizing the planar Josephson junction scheme.

Two experiments attempting to realize this planar proposal are worth mentioning at this point. Ren et al [52] used a HgTe quantum well as the 2DEG (which has the advantage of very strong spin–orbit coupling), and thermally evaporated thin-film Al on top of it. Fornieri et al [53] used InAs as the 2DEG (which has the advantage of high mobility), with a thin Al layer epitaxially grown on top. The two experiments reported zero-bias conductance peaks consistent with topological superconductivity. Curiously, the experiments were performed with different material platforms and at quite different length regimes (narrow vs. wide junction, narrow vs. wide SC leads compared to the SC coherence length), but they both showed encouraging signatures of MZMs.

As a closing remark, we touch upon a delicate issue which is of primary importance in experiments. The model of [35, 36] indeed predicts a topological phase, but its topological gap is only about a third of the induced SC gap, at best. The culprit turns out to be long semiclassical trajectories that do not scatter off SCs, and thus lead to a small gap at finite k_x . One way to get rid of these trajectories is impurities, which break the trajectories and make them scatter onto the SCs too. Indeed, it was found that the right amount of disorder may increase the gap and shorten the Majorana localization length [54]. A different approach is changing the geometry of the junction such that these trajectories naturally scatter, for example by using a zig-zag shape [55]. This was found to confine the MZM wavefunction more compared to the straight junction case.

The original models of semiconducting nanowires [33, 34] included the effect of the magnetic field only via the Zeeman splitting. In other words, the nanowires were treated as truly one-dimensional objects, so orbital effects were neglected. In reality, however, nanowires have a finite cross section so flux can be threaded through them. This important effect is at the core of the full-shell nanowires scheme proposed by Vaitiekėnas et al [57].

Consider a cylindrical nanowire (e.g. InAs) fully coated by a superconductor (e.g. Al), see figure 4. The spin–orbit coupling in the nanowire arises from local breaking of inversion symmetry, and crucially the effective electric field points in the radial direction $\hat{r}$. The continuum normal-state Hamiltonian is then

$\begin{equation} H_0 = \frac{\left(\vec{p}-e\vec{A}\right)^2}{2m} - \mu + \alpha\hat{r}\cdot\left[ \vec{\sigma}\times \left(\vec{p}-e\vec{A}\right) \right]. \end{equation} \tag{ 13 }$

Here $\vec{A}$ is the vector potential, α is the Rashba spin–orbit coupling strength, and $\vec{\sigma}$ is the vector of Pauli matrices acting in spin space. We assume a magnetic field $\vec{B} = B\hat{z}$ is applied along the nanowire's axis, so a reasonable gauge choice is $\vec{A} = \frac{1}{2}\vec{B}\times\vec{r}$.

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On top of this normal-state Hamiltonian, we add the superconducting pairing, which is modulated in space due to the presence of the magnetic field: $\Delta\left(\vec{r}\right) = \Delta e^{-in\phi}$, where φ is the angular coordinate. The winding n is strictly an integer, due to the wavefunction being single valued. An important observation of Vaitiekėnas et al [57] is that the BdG Hamiltonian commutes with the generalized angular momentum operator

$\begin{equation} J_z = -i\partial_{\phi} + \frac{1}{2}\sigma_z + \frac{n}{2}\tau_z. \end{equation} \tag{ 14 }$

The first two terms are not surprising—they are nothing but the orbital and spin angular momenta. The last term, however, is special: the phase winding alters the angular momentum and adds a contribution that is opposite for particles and holes. This is yet another manifestation of the notion (that has hopefully become more transparent at this point) that SC phase gradient can be thought of as a special type of orbital field.

Inspecting equation (14), we see that the eigenvalue m_J of the orbital angular momentum $-i\partial_{\phi}$ takes on integer values of odd n and half-integer values for even n. Since the particle-hole operator (which by necessity anti-commutes with the BdG Hamiltonian) relates a state with energy $E,m_J$ to a state with $-E,-m_J$, it is evident that at the $m_J = 0$ sector there is hope for non-degenerate Majorana zero modes. That is because the particle-hole symmetry at $E = 0,m_J = 0$ relates a state with itself and therefore does not add any degeneracy. It is therefore worthwhile to examine $m_J = 0$, and that can only be done at odd n for this system. The conclusion is that an odd winding—one vortex at least—is needed for a topological transition.

Following this line of reasoning, the authors of [57] then apply a unitary transformation to the BdG Hamiltonian to bring it to a form similar to the original nanowire Hamiltonian [33, 34], see equation (3). We do not repeat the full expressions here, but the key point is that at $m_J = 0$ the model maps exactly to the Majorana nanowire Hamiltonian, with a chemical potential shifted by the flux and the spin–orbit coupling, and an emergent Zeeman field,

$\begin{equation} E_\textrm{Z} = \left(n-\frac{\Phi}{\Phi_0}\right) \left( \frac{1}{4mR^2} + \frac{\alpha}{2R} \right), \end{equation} \tag{ 15 }$

where R is the cylinder's radius (assuming a thin superconductor) and $\Phi = \pi R^2 B$ is the flux threaded through the cylinder. It is thus straightforward to derive the criterion for a topological phase in this description, and indeed the authors found large topological regions in parameter space. We stress again that no Zeeman field was assumed to be present in the original Hamiltonian, and it is indeed the SC phase winding that gives rise to the effective Zeeman field.

The relevant magnetic field scale for a topological transition is determined by $\Phi\approx\Phi_0$, i.e. by the geometry of the nanowire. Notice again that this is a purely orbital effect (no g factors involved). In practice, for the Al-coated InAs nanowire studied in [57], this field scale is around 100 mT (significantly lower than the field scale for a Zeeman-driven transition in proximitized InAs nanowires [58, 59]). The zero-bias peaks, which may imply the existence of Majorana zero modes, are observed experimentally as additional features on top of the Little-Parks oscillations.

We note that a similar phenomenon of orbitally driven topological superconductivity has been pointed out in carbon nanotubes [60]. These are sheets of graphene rolled up to form quasi-1D tubes, and due to their finite diameter, they can also capture magnetic flux. Since they are made entirely of carbon atoms, carbon nanotubes have an extremely small g factor, and therefore the orbital effect can in fact be the dominant one. However, the band structure and spin–orbit coupling in carbon nanotubes are markedly different compared to semiconducting nanowires such as InAs, making the relevant physics quite different. We note that Zeeman-based schemes in CNTs have also been proposed [61, 62], but they require very large magnetic fields.

The realization that phase bias can be an alternative to applying a Zeeman field was also adopted in the context of planar Josephson junctions (see section 3.2 and [35, 36]) by Melo et al [63]. Recall that the original proposal for topological superconductivity [35, 36] already has phase bias, but only in one direction (y in the notation of figure 3). Melo et al [63] proposed to extend this to a continuous phase gradient along the junction (x in the notation of figure 3), thereby potentially eliminating the need for a Zeeman field in this direction. The need for a phase gradient in both dimensions becomes clear when considering $k_x = 0$. In the absence of phase gradient, a y-dependent gauge transformation removes the spin–orbit coupling. In the absence of a Zeeman field in the x direction, this makes all states doubly degenerate, thus prohibiting a transition into the topologically non-trivial phase.

The first step taken by Melo et al is applying supercurrents to the two SCs, generally with different gradients:

$\begin{equation} \phi(y) = \left\{\begin{array}{ll} \exp\left(2\pi i x/\lambda_\textrm{L}\right) & \textrm{for}\ y \lt 0\\ \exp\left(-2\pi i x/\lambda_\textrm{R}\right) & \textrm{for}\ y \gt 0 \end{array}.\right. \end{equation} \tag{ 16 }$

Notice that the effective unit cell of the junction is now enlarged: it is the least common multiple of λ_L and λ_R. While these supercurrents lift the degeneracy at $k_x = 0$, the spectrum turns out to be gapless due to a gap closing at finite k_x . This gap closing is a result of a symmetry dubbed 'charge-momentum parity', which reflects the fact that Andreev reflections in this system are always accompanied by a momentum shift of $\pm2\pi/\lambda_\textrm{L}$ or $\pm2\pi/\lambda_\textrm{R}$. This symmetry, ${\cal O} = (-1)^n \tau_z$ (where n is the number of unit cells in reciprocal space), which commutes with the Hamiltonian and anti-commutes with the particle-hole operator, prohibits a gapped topological phase.

Several mechanisms are proposed to break the charge-momentum parity:

• (a)  
  Adding a periodic potential, which scatters states of different wavevectors inside the electron and hole sectors separately.
• (b)  
  Deforming the junction to a non-straight shape such as zig-zag. Here, the geometry makes the Andreev reflection change k_x in a way that breaks $\cal O$.
• (c)  
  Adding a third SC without supercurrent in the middle of the junction (creating a SNSNS junction). The additional SC couples states at k_x and $-k_x$ without any momentum transfer.

Interestingly, in cases (a) and (b), modulations commensurate with λ_L and λ_R are favorable for breaking ${\cal O}$ and opening a topological gap, whereas incommensurate modulations facilitate a gapless state. This is because in the commensurate case—and always in case (c) above if $\lambda_\textrm{R} = \lambda_\textrm{L}$—inversion symmetry is maintained. When inversion symmetry is broken, the spectrum gets tilted and can easily become gapless, and thus retaining inversion symmetry is beneficial in this setup.

An important lesson to learn from [63] is that SC phase winding can be a sufficient time-reversal-symmetry breaking mechanism, but the directions matter a lot. If time-reversal is broken in such a way that the effective Zeeman field is parallel to the spin–orbit coupling, the spectrum simply becomes gapless. In fact, even if the SC phase changes in all directions, one still needs clever tricks to avoid gap closings.

A different approach to phase-only Majorana realization, introduced in [64, 65], utilizes the concept of a discrete vortex (see section 2) in conjunction with spin–orbit coupling. Here we review the basic idea, and two practical routes for implementation.

The analysis begins with a spin-orbit-coupled quantum ring, where each site is coupled to a superconductor which has a different phase, see figure 5. Form [45] we know that three different superconducting phases are necessary to have a zero-energy Andreev state. Since we will be interested in forming zero-energy states, the minimal number of sites in the ring is three.

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The Hamiltonian describing the ring is [64, 65]

$\begin{align} H = &-\sum_{n = 1}^{3} \mu c^{\dagger}_n c_n \nonumber \\ &+ \sum_{n = 1}^{3}\left( -c^{\dagger}_n te^{i\lambda_n \sigma_z} c_{n+1} + \Delta e^{i\phi_n} c^{\dagger}_{n\uparrow} c^{\dagger}_{n\downarrow} + \textrm{H.c.} \right). \end{align} \tag{ 17 }$

Here n enumerates the three sites of the ring, $c_n = \left( c_{n\uparrow}, c_{n\downarrow} \right)^\textrm{T}$ and the phases λ_n characterize the spin–orbit coupling strength at each bond. The reader might be concerned about the SOC being always related with the σ_z operator and not the other spin operators; a seemingly more appropriate operator might be $\sigma_{\theta} = -\sigma_{x}\sin\theta+\sigma_{y}\cos\theta$, where θ is the angular polar coordinate. The two representations are actually related by the local transformation $\mathcal{U} = \exp\left[\frac{i}{2}\left(\theta+\frac{\pi}{2}\right)\sigma_{z}\right]$ (see also [57]): $\mathcal{U}\sigma_{\theta}\mathcal{U}^{\dagger} = \sigma_{x}$ and σ_x may easily be rotated into σ_z . The SC part of the Hamiltonian, being a spin-singlet term, is invariant under such a rotation, thereby justifying the use of the simpler SOC term appearing in equation (17), by essentially redefining $c_n\to {\cal U}^{\dagger} c_n{\cal U}$.

Let us first consider the simple case of uniform SOC, $\lambda_1 = \lambda_2 = \lambda_3\equiv\lambda$, and equal SC phase differences, $\phi_n = \frac{2\pi}{3}mn$ where $m = \pm1$. In this case the model has translation invariance and it is conveniently analyzed in Fourier space, using the momentum around the ring k as a conserved quantum number. The Bloch-BdG Hamiltonian is

$\begin{align} \mathcal{H}\left(k\right) = \left( \begin{array}{cccc} \epsilon_{k\uparrow} & 0 & \Delta & 0\\ 0 & \epsilon_{k\downarrow} & 0 & \Delta\\ \Delta & 0 & -\epsilon_{-k\downarrow} & 0\\ 0 & \Delta & 0 & -\epsilon_{-k\uparrow} \end{array} \right), \end{align} \tag{ 18 }$

where $\epsilon_{k\sigma} = -\mu+2t\cos\left(k+\frac{\pi m}{3}+\lambda\sigma\right)$ and as usual the Nambu spinor takes the form $\vec{\Psi}_{k} = \left(c_{k\uparrow},c_{k\downarrow},c_{-k\downarrow}^{\dagger},-c_{-k\uparrow}^{\dagger}\right)^\textrm{T}$. To find zero-energy states, we demand that for some value of k,

$\begin{equation} \det\mathcal{H}\left(k\right) = \left(\Delta^{2}+\epsilon_{k\uparrow}\epsilon_{-k\downarrow}\right)\left(\Delta^{2}+\epsilon_{k\downarrow}\epsilon_{-k\uparrow}\right) = 0. \end{equation} \tag{ 19 }$

There are many ways to fulfil this equation. Let us focus on a case that simplifies the calculation: $\epsilon_{k\sigma} = -\epsilon_{-k,-\sigma} = \Delta$ for some specific choice of $k,\sigma$. This leads to two equations (see [65] for additional details), that can be further simplified by setting µ = 0. This leads to the requirement $\Delta = \pm\sqrt{3}t$, and to a requirement on the SOC parameter (for m = 1):

$\begin{align} k + \lambda\sigma + \frac{\pi m}{3} = \pm\frac{\pi}{6} + 2\pi N, \end{align} \tag{ 20 }$

where N is an integer. For example, the choice k = 0, $\sigma = \uparrow$ leads to $\lambda = -\frac{\pi}{2}+2\pi N$. We therefore conclude that it is possible to induce zero-energy states, which are delocalized Majorana states, using just three phases. Notice the interplay between λ, the Aharonov–Casher phase [51], and m, the SC phase winding.

The analysis may be generalized to a ring without translation invariance [64]. In this case, the determinant of the full BdG Hamiltonian (17) may be written explicitly:

$\begin{align} \det H& = 6\mu^{2}t^{2}\left(\Delta^{2}+\mu^{2}\right) -\left(\Delta^{2}+\mu^{2}\right)^{3} \nonumber\\ &\quad -3t^{4}\left(\Delta^{2}+3\mu^{2}\right) -2f\Delta^{2}t^{2}\left(\Delta^{2}+\mu^{2}+t^{2}\right) \nonumber \\ &\quad -4\mu t^{3}\Lambda\left(f\Delta^{2}-\mu^{2}+3t^{2}\right) -4t^{6}\Lambda^2, \end{align} \tag{ 21 }$

where $f = \cos\left(\phi_{1}-\phi_{2}\right)+\cos\left(\phi_{2}-\phi_{3}\right)+\cos\left(\phi_{3}-\phi_{1}\right)$ and $\Lambda = \cos\left(\lambda_1+\lambda_2+\lambda_3\right)$. To find zero-energy states, we look for solutions of the equation $\textrm{det}H = 0$; notice that these only depend on the total spin-orbit phase (characterized by Λ), and the SC phases enter through a single gauge-invariant parameter f.

Treating equation (21) as a quadratic equation for Λ, it is readily understood that regardless the values of $t,\,\mu,$ and Δ, real solutions of Λ are possible only when $f\leqslant -1$. It can readily be shown [64] that f is directly related to phase winding: a discrete vortex is formed if and only if $f\leqslant-1$. Therefore, we find that phase winding is a necessary condition for a zero-energy state. Continuing the analysis of the quadratic equation, one can then find a set of optimal values of $\mu/t,\Delta/t,\Lambda$, such that $f\leqslant-1$ becomes a sufficient condition: for these values, any $-3/2\leqslant f\leqslant-1$ will induce a zero-energy state. The manifold in this three-dimensional parameter space turns out to be the circle $\Delta = \sqrt{t^2-\mu^2}$, $\Lambda = \mu/t$, and on this circle we say that $f_\textrm{crit} = -1$. This criterion may be translated to continuum parameters, leading to the rule of thumb $L\Delta\approx\alpha$, where L is the typical length of the system and α is the spin–orbit coupling constant. Moving away from this optimal circle reduces $f_\textrm{crit}$ (notice that smaller $f_\textrm{crit}$ is a more stringent condition on the phases; the minimal value f may take is $-3/2$ and it corresponds to a perfect vortex).

At this point, we would like to extend this ring to a 1D system, building on the observations above. One can think of two ways to do that, which can be regarded as different ways to concatenate rings. The first way we describe amounts to coupling the rings out of the plane, forming a cylinder-like structure, as described in [64]. The other way, which is explored in [65] is repeating the rings in the plane.

Consider promoting each site of the ring of figure 5 to be a wire, as depicted in figure 6(a). The ring's Hamiltonian equation (17) is then understood as the $k_{\parallel} = 0$ Hamiltonian of the coupled wires system. Therefore, the zero-energy states of this Hamiltonian correspond to topological phase transitions. We can use the form of the determinant, equation (21), to understand the topological phase diagram. From our previous arguments we know that phase winding, or $f\leqslant-1$, is a necessary condition for a topological phase, and we also know when this becomes a sufficient condition. On the optimal manifold we found, any $-3/2\leqslant f\lt-1$ will drive the system into the topological phase, and the phase diagram will assume the form of triangles as in the Fu–Kane model (see [42] and figure 1). Away from this optimal manifold, the phase boundaries get distorted: they are no longer straight lines, but rather contours of constant f (see figure 6(c)).

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An important consideration we need to account for is the topological gap. The system can only be topological if it has a finite gap, and the larger the gap the better: it provides the protection of the MZMs and determines their localization length. One may expect the gap to be largest when the phases form a perfect vortex (${\,\,f = -3/2}$). However, it turns out that this is not the case: for such a configuration, the system has C₃ rotation symmetry, which manifests itself as a gap closing at finite $k_{\parallel}$. Therefore, besides optimizing the topological region in parameter space, one should also be concerned with the non-trivial evolution of the gap. The optimal topological gap is numerically found to be about a third of the induced SC gap [64].

This non-planar construction may be extended to a more realistic setup. Consider smoothly deforming the cylinder into a rectangular box, which we model as a bilayer, see figure 6(b). The cylindrical geometry is manifested by a different SOC field in each layer. Another way to make this transition is to imagine turning off the periodic boundary conditions, so we are left with just three wires in the plane. The SOC cannot contribute in this scenario—we need to engineer a way for it to not cancel out. This is done by introducing an additional band in each wire, with a different SOC amplitude. Electrons can then travel in closed trajectories that capture a non-zero Aharonov–Casher phase; for example, the trajectory can go left-to-right in the 'top' layer, then hop down, return right-to-left, and finally hop back up. The advantage of this geometry is its experimental accessibility by means of simple semiconducting quantum wells, and indeed it has a very similar phase diagram to the cylinder model [64].

The second approach to this phase-only setup is concatenating the rings of figure 5 in the plane. A conceptually simple construction goes as follows: we take two rings and tune each of them to have zero-energy states. For simplicity, this can be done using the assumption of a perfect vortex (the details of the derivation in this case are given in [65], but notice that it is not essential to assume a perfect vortex). At this point, we have a total of four MZMs, two localized in each ring. Our goal is to turn on coupling between the rings such that two of these states are gapped out, and we are left with one MZM per ring. It turns out that to achieve this, the rings must be connected via more than one link (the physical reason for this is that eliminating two MZMs while leaving the other two intact is only possible via interference). The coupling may be chosen such that this elimination is perfect, leading to a so-called 'sweet spot', analogous to the Kitaev chain in the $t = \Delta$, µ = 0 limit. It is now possible to repeat this two-rings unit cell as many times a one desires, and regardless of how the unit cells are coupled, we will end up with one MZM localized at each end of the system. The conclusion is that phase bias can be used to drive the system into the most localized topological regime possible.

As in the case of the non-planar model, this planar construction may also be adopted to a more experimentally feasible setup [65]. A proposal for such a setup is shown in figure 7. The idea is forcing electrons to go in closed trajectories that pick up an Aharonov–Casher phase (due to Rashba SOC in a 2DEG) and are also sensitive to the phase bias. Due to the inherent asymmetry between the three phases in this proposal, it does not suffer from the problem of gap closing in the presence of a perfect vortex. Other than that, its topological phase diagram is very similar to that of the non-planar model.

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