Abstract
Recent experimental efforts towards the detection of Majorana bound states have focused on creating the conditions for topological superconductivity. Here we demonstrate an alternative route, which achieves fully localised zeroenergy Majorana bound states when a topologically trivial superconductor is strongly coupled to a helical normal region. Such a junction can be experimentally realised by e.g. proximitizing a finite section of a nanowire with spinorbit coupling and combining electrostatic depletion and a Zeeman field to drive the nonproximitized (normal) portion into a helical phase. Majorana zero modes emerge in such an open system without finetuning as a result of chargeconjugation symmetry and can be ultimately linked to the existence of ‘exceptional points’ (EPs) in parameter space, where two quasibound Andreev levels bifurcate into two quasibound Majorana zero modes. After the EP, one of the latter becomes nondecaying as the junction approaches perfect Andreev reflection, thus resulting in a Majorana dark state (MDS) localised at the NS junction. We show that MDSs exhibit the full range of properties associated to conventional closedsystem Majorana bound states (zeroenergy, selfconjugation, 4πJosephson effect and nonAbelian braiding statistics), while not requiring topological superconductivity.
Introduction
The emergence of topologically protected Majorana zero modes in topological superconductors has recently entered the spotlight of condensed matter research^{1,2,3,4,5}. One of the main reasons is the remarkable prediction that such Majorana bound states (MBSs), also known as Majorana zero modes, should obey nonAbelian braiding statistics^{6,7}, much like the 5/2 states in the fractional Hall effect, without requiring manybody correlations. It has been argued that the successful generation, detection and manipulation of MBSs would open the possibility of practical topologically protected quantum computation^{8,9}. Despite impressive experimental progress^{10,11,12,13,14,15,16,17,18,19}, such ambitious goals have still not been conclusively achieved.
A number of practical proposals have been put forward aiming to generate the conditions for the spontaneous emergence of robust MBSs in real devices. Some of the most studied ones are based on proximitizing topological insulators^{20} or semiconductor nanowires^{21,22}. The core challenge in all these proposals has been to artificially synthesise a topologically nontrivial superconductor with a welldefined and robust topological gap^{23}. The bulkboundary correspondence principle dictates that the superconductor surface is then host to topologically protected MBSs. Creating a topological gap is arguably the main practical difficulty of such proposals, particularly since topological superconductors are rather sensitive to disorder.
In this work we demonstrate an alternative scheme for the creation of MBSs that does not require topological superconductivity at all. The possibility of engineering Majoranas in topologically trivial setups has been studied in other contexts before. It has been shown, for example, that topological excitations and MBSs in particular, may arise in trivial superconductors under adequate external driving^{24,25}, similarly to the mechanism behind Floquet topological insulators^{26}. Also, coldatom systems with specifically engineered dissipation^{27,28} may relax into a topologically nontrivial steady state that are host to dark states at zero energy with Majorana properties. Our approach is implemented in a solid state setup and is based on proximitized semiconductor nanowires. In its topologically trivial regime, such a wire will not generate MBSs when terminated with vacuum (i.e. at a closed boundary). Its spectrum is instead a set of Bogoliubov quasiparticles that can be seen as pairs of Majoranas hybridized to finite energy. By creating a sufficiently transparent normalsuperconductor (NS) junction at one end of the wire, we create a different kind of open boundary, to which the bulkboundary correspondence principle does not apply. Such a hightransparency junction can be fabricated by proximitizing only one half of a pristine semiconducting nanowire (Fig. 1). We demonstrate that, as one tunes the normal side into a helical (halfmetallic) regime via a parallel Zeeman field, one Majorana pair becomes decoupled into two zero energy resonances. One of which is subsequently removed into the reservoir, leaving behind a stable Majorana ‘dark state’ (MDS) at the NS junction without requiring a nontrivial superconductivity. (A dark state here is defined as a bound state that despite having an energy embedded in a continuum of delocalized excitations is orthogonal to them and, therefore, nondecaying.)
The emergence of these MDSs cannot be described using the conventional band topology language, but rather needs to be understood in the context of open quantum systems. Unlike in closed systems, eigenstates in open systems decay with time, as the state leaks into the reservoir. Hence, their energies ε_{p} = E_{p} − iΓ_{p} are no longer real, but have a negative imaginary part that represents this decay rate Γ_{p}. Such a complex spectrum is sometimes modelled by a nonHermitian Hamiltonian. A more precise and general description is obtained by considering the analytic continuation of the scattering matrix S(ω), where the energy ω of incoming states from the reservoir is allowed to extend into the lower complex halfplane. The analogous to the real eigenvalues of the closed system then becomes the poles of S(ω) for the open system. In NS junctions, all such poles come in pairs ±E_{p} − iΓ_{p} for all poles with nonzero real part E_{p}, due to the chargeconjugation symmetry of the Nambu representation^{29,30}. In this sense, poles with zero real part E_{p} = 0 are special, as need not come in pairs come in pairs. Their total number Z (which by convention excludes ‘buried poles’, with zero real and imaginary part) has a very important meaning in open NS junctions and defines the analogue of band topology of a closed quantum system. Indeed, it has been shown that the topology of the scattering matrix in quasi1D NS junctions is classified by the invariant ν = Z, mod,2 i.e. the parity of the number of poles with zero real energy, with ν = 1 signalling an open system with nontrivial topology from the point of view of scattering^{29,30}.
In terms of its Smatrix poles, the topological transitions of an NS junction follow a characteristic pattern. Consider a trivial Smatrix (ν = 0). As a given parameter of the system is varied, a pair of poles ε_{p} = ±E_{p} − iΓ_{p} first approach each other and become degenerate at the imaginary axis, E_{p} = 0. This degeneracy is the opensystem counterpart of a band inversion in a closed system and is known as an exceptional point (EP)^{31,32,33,34,35}. It differs from a closedsystem degeneracy in that the corresponding eigenstates do not remain orthogonal, but rather coalesce into one. Exceptional points have been extensively studied in photonics where they have been shown to give rise to novel phenomena unique to open systems^{36,37,38,39,40,41,42,43}. Their implications in electronic systems, however, have been seldom discussed^{44,45}.
After the exceptional point, the two degenerate poles branch along the imaginary axis and their decay rates bifurcate into different values Γ_{0} < Γ_{1} (see Fig. 2d for an example). The exceptional point thus involves a change of Z by 2, but ν = 0 remains unchanged. If Γ_{0} evolves towards zero (or close enough to zero for all practical purposes), it is said that the corresponding pole is buried and it is excluded from the N count, effectively signalling a change of topology ν = 1. Crucially, the existence of a buried pole implies the existence of a zeroenergy nondecaying (dark) state somewhere in the system with Majorana properties. In this sense, Smatrix topology is a true generalization of the bandstructure topology of closed systems and has the same implications in terms of topologically protected excitations, albeit in the context of open systems. It is also closely linked to the existence of an exceptional point in the system that occurs before the pole burying, in the trivial ν = 0 phase.
Here we show that, while at weak couplings between the normal environment and the superconductor a nontrivial Smatrix implies that the superconductor is also nontrivial in isolation, this is not the case at strong couplings. In specific but experimentally relevant conditions, a sufficiently transparent junction between a normal metal and a trivial superconductor has a nontrivial Smatrix with ν = 1 and is thus host to a Majorana dark state. The required conditions are: (1) the system should have a finite spinorbit coupling at the contact, (2) the normal part of the junction should be sufficiently depleted (small Fermi energy μ_{N}) and polarised by a Zeeman field B into a helical halfmetallic phase B > μ_{N}, (3) the normal transmission of the junction should be close to one and (4) the trivial superconductor should be in the Andreev limit Δ ≪ μ_{S}, where Δ is the superconducting gap and μ_{S} is its Fermi energy. The rationale of these conditions is to achieve good Andreev reflection of helical carriers from the normal side, which generates a MDS strongly localized at the junction. The intuitive mechanism behind the process is as follows. When isolated, the trivial superconducting wire is host to a Majorana pair at each end, which is strongly hybridized into a fermionic Bogoliubov state, with real energies ±E_{p}. As the contact is opened onto a helical wire (which has a single decay channel), one (and only one) of the two Majoranas escapes into the reservoir (blue state in Fig. 1), leaving behind the orthogonal Majorana as a dark state (red), pinned at zero energy since it no longer overlaps with the escaped Majorana. This process takes place formally by crossing an exceptional point bifurcation of the ±E_{p} poles into poles iΓ_{0,1} on the imaginary axis. As the conditions above are fulfilled, the zero energy dark state becomes truly nondecaying Γ_{0} → 0. Deviations from these conditions result in a residual decay rate Γ_{0}. The dark state is then a sharp Majorana resonance centered at zero, but with Majorana properties surviving at times shorter than τ_{0} = 1/Γ_{0}.
We analyse the generation of a MDSs in two different models. First we consider the problem of the open multimode Kitaev wire and see how the above scenario plays out in this simple model. Then we consider a more realistic model of an NS contact in a proximitized semiconducting wire. We describe the experimental signatures associated to the MDS. We also analyze its properties of the dark state to demonstrate it indeed shares all the characteristics of a Majorana bound state from a closed topological system, including selfconjugation, locality and neutrality and the appearance of uniform charge oscillations, 4π Josephson effect and nonAbelian braiding when combined with another MDS. We show that the residual decay rate of the MDS in nonideal conditions depend exponentially with junction lengthscales, just like the Majorana splitting in closed topological superconductors.
Results
Exceptional points and Majorana dark states in the open Kitaev model
We first study the emergence of stable zeroenergy dark states in an open toy model, in preparation for the more realistic calculation presented later for proximitized semiconducting wires. We consider a natural multimode Dclass extension^{46,47} of the original Kitaev model^{6}. It is a finite length (i.e. closed) quasi1D chain of spinless fermions with a p_{x} + ip_{y} superconducting pairing with interesting topological properties. It may be written as a nearestneighbour tightbinding Hamiltonian in an L × N square lattice,
The hopping amplitude is t, are integer site indices and μ is the chemical potential. The p_{x} + ip_{y} pairing is implemented by and . For N > 1, t > Δ and finite length L ≫ 1 (wire terminated by vacuum), this model exhibits an evenodd effect in its spectrum^{46} as μ is varied, see Fig. 2b. Odd (nontrivial) phases develop a single zeroenergy MBS at each end of the wire (in red, Majorana number ν = 1), while even (trivial) phases have none (ν = 0). These MBSs arise as a result of the bulkboundary correspondence principle when the bulk bandstructure of the wire becomes topologically nontrivial, with ν as the topological invariant^{47,48,49}.
We now analyse the behaviour of the spectrum with a different type of boundary. Instead of vacuum we couple the wire to a normal singlemode reservoir at each end. This is a very different boundary (it is gapless), to which the bulkboundary correspondence principle is not applicable. We will show how a strong enough coupling to this specific singlemode reservoir may allow certain topologically trivial phases (ν = 0, no MBSs with vacuum termination) to develop stable bound states at the contacts, or dark states pinned to zero energy and closely related to the MBSs of nontrivial phases.
For concreteness we consider the N = 2 Kitaev wire, which is the simplest that manifests this phenomenon. The wire is coupled to singlemode (N = 1) normal reservoirs at each end, as shown in Fig. 2a. Its effect is modelled through a nonHermitian selfenergy Σ_{res} on one end site, which results in a nonHermitian effective Hamiltonian for the open multimode Kitaev wire
The local decay rate Γ_{res} depends on the (constant) local density of states of the reservoir at the contact ρ_{res} and the hopping amplitude t′ into the reservoir, Γ_{res} = πt′^{2}ρ_{res}. The shift ReΣ_{res} is neglected for the moment.
The eigenvalues of are complex ε_{p} = E_{p} − iΓ_{p} and represent the real energy E_{p} and decay rate Γ_{p} into the reservoirs of quasibound states in the wire. They are also the poles of the scattering matrix S(ω) from the reservoir for complex ω. Causality implies Im ε_{p} < 0, while the chargeconjugation symmetry of the Bogoliubovde Gennes description guarantees that if ε_{p} is an eigenvalue, then is also. This symmetry classifies eigenvalues in conjugate pairs, unless they lie exactly on the imaginary axis, in which case they may have no partner. As summarised in the introduction, the number of such lone, purely imaginary (or zero) eigenvalues per edge is denoted by Z and carries a profound physical significance related to the topology of the scattering matrix at that contact. The number Z allowed Pikulin and Nazarov^{29,30} to classify the scattering matrices S(ω) of generic NS junctions into trivial (even Z) and nontrivial (odd Z) topological classes and connect it to bulk topological invariants by ν = Z, mod,2. Z is moreover a robust quantity. An unpaired quasibound state with ε_{p} = −iΓ_{p} cannot acquire a finite real energy E_{p} through any small perturbation, just change its decay rate Γ_{p}. This robust pinning to zero real energy, protected by chargeconjugation symmetry, is reminiscent of the topologically protected zero energy pinning of MBSs in closed topological superconductors.
The evolution of the complex ε_{p} as a function of Γ_{res} for μ = 0 is shown in Fig. 2(c,d) and the phase diagram of the model in the full (μ, Γ_{res}) plane is shown in Fig. 2e. The trivial ν = Z = 0 phase at Γ_{res} = 0 around μ = 0 is characterized by two subgap nonzero eigenvalues resulting from the hybridization of two MBSs at each edge [black curves in panel (b)]. As Γ_{res} is increased, this trivial phase experiences a transition into Z = 2 through the fusion of these two levels. This is an instance of an ‘exceptional point’, a generic feature of open systems, or in general of nonHermitian matrices with certain symmetries^{34,45}, at which two complex eigenvalues become degenerate and bifurcate in the complex plane. One such bifurcation is shown in panels (c,d). At the exceptional point (here Γ_{res} ≈ 0.55t for μ = 0), the corresponding eigenstates coalesce [their overlap reaches one, dotdashed green curve in panel (c)], in contrast to the Hermitian case where degenerate eigenstates remain orthogonal.
After crossing the exceptional point, the two hybridized Majoranas become exact zero modes, albeit with different lifetimes, without any fine tuning. One of the two eigenvalues at each contact gradually approaches the origin [red in panel (d)], i.e. its imaginary part also approaches zero. We denote this imaginary part, or decay rate, by Γ_{0}. Note that Γ_{0} is not of order Γ_{res}. In fact Γ_{0} is suppressed as Γ_{res} increases, see panels (c,e). As a result, the corresponding state evolves into a longlived zeroenergy resonance localised at an increasingly transparent contact. The suppression of its decay Γ_{0} is a result of the specific singlemode nature of the reservoirs considered here and is not related to any change in the bulk topology, which remains trivial ν = 0.
Physically, the above bifurcation can be understood as follows. In the N = 2 trivial phase around μ = 0, the Kitaev wire actually hosts two Majorana bound states at each end, initially at zero energy, that owing to their strong overlap hybridize into finite energy (thick black line in Fig. 2b). These two states are schematically represented by the red and blue circles in Fig. 2a. As the wire is opened to the singlemode reservoir, only one combination of these two states (blue circle) can scape across the transparent junction into the reservoir while its overlap with the orthogonal combination (red circle), which remains localized, is suppressed. As a result, the decay rates of the two states bifurcate at the exceptional point while their real part becomes exactly zero. Thus, the fact that the reservoir has a single mode into which only one of the two Majoranas may decay is essential in order to yield one decoupled dark state at a transparent junction. Here, transparent should be understood as perfect Andreev reflection probability. This implies perfect delocalisation of one Majorana and a decoupled zeroenergy dark state. To have a perfect Andreev reflection, however, it is not enough to have a transparent contact in the normal phase (Δ = 0). It is also necessary that the pairing is a small perturbation to the normal phase (Δ ≪ t, Andreev limit). Otherwise a nonzero normal reflection will arise at the NS junction, leading to a finite residual decay Γ_{0}. As we approach the Andreev limit, the suppression of the residual decay rate Γ_{0} at large Γ_{res} is very fast, decreasing exponentially with the ratio t/Δ (not shown). Hence, the Majorana resonance quickly becomes a proper MBS for t, Γ_{res} ≫ Δ.
It has been shown that realistic models for an isolated, Zeemanpolarised, proximitized 1D Rashba wire belongs to the same topological class as the Kitaev model^{1}. Such models also develop a ν = 1 topological phase with MBSs for strong enough Zeeman fields, while they remain trivial ν = 0 at low fields. In the following we study the emergence at exceptional points of zero energy dark states with Majorana properties in open topologically trivial Rashba wires.
Majorana dark states in a proximitized Rashba wire
In recent years, experimental progress has been reported towards the detection of MBS in Rashba nanowires^{10,11,12,13,14,15,16}. These efforts were in large part stimulated by the prediction by Lutchyn et al.^{21} and Oreg et al.^{22} that these type of systems would undergo a topological transition into an effective pwave superconducting phase when a Zeeman field B parallel to the wire exceeds a critical value B_{c}. We now consider models relevant to singlemode Rashba wires and demonstrate the formation of MDSs for B < B_{c}.
Following refs 21,22, we model a thin proximitized Rashba nanowire under a Zeeman field by a spinful 1D tightbinding chain,
The parameters of the model are the chemical potential measured from depletion μ_{S}, the hopping , where m^{*} is the effective mass and a_{0} is the lattice spacing, the induced pairing Δ, the SO hopping , where α_{SO} = ^{2}/(m^{*}λ_{SO}) is the SO coupling and λ_{SO} is the SO length and the Zeeman field , where g is the gfactor and is the magnetic field along the wire. In what follows we present simulations with parameters corresponding to an InSb proximitized wire^{10} (Δ = 0.25 meV, α_{SO} = 20 meV nm, m^{*} = 0.015m_{e}, g = 40). Like for the Kitaev wire, we will consider both an isolated proximitized wire of finite length and an open NS contact between proximitized and nonproximitized sections of a nanowire, Fig. 1. The latter is assumed infinite (see the supplemental information for finite length effects) and is modelled by the same Hamiltonian, albeit with Δ = 0 and a μ_{N} in place of μ_{S}. The normalstate average transparency per mode T_{N} of the contact is physically controlled by a electrostatic gating in an actual device and is modelled here either by a hopping t′ ≤ t across the contact, or by a spatial interpolation between μ_{N} and μ_{S} across a certain contact length L_{C} determined by the distance of the wire to the depletion gate (Δ is always abrupt, see Supplementary Material). Note that if the density of defects in the wire is small, T_{N} ~ 1 − exp(−L_{C}/λ), with λ a lengthscale of the order of the average Fermi wavelength.
Topologically, the isolated nanowire belongs, for finite B, to the same onedimensional Dclass as the multimode Kitaev wire of the preceding section. For B smaller than a critical , The nanowire is trivial (ν = Z = 0). As B exceeds B_{c}, the topological invariant becomes nontrivial (ν = 1) through a band inversion, see Fig. 3a, much like the transition at μ ≈ t in Fig. 2b, with the peculiarity that the two hybridized Majoranas in the trivial B < B_{c} phase are not deep inside the gap, but at the band edge. As μ_{S} grows, B_{c} quickly becomes unrealistically large and the nanowire remains trivial for all reasonable fields, Fig. 3b. (A large μ_{S}, incidentally, is the natural experimental regime, since the superconductor will typically transfer charge to the proximitized section of the wire that is difficult to deplete due to screening.)
As the nanowire contains a nonproximitized normal section, Fig. 1, a phenomenology similar to the open N = 2 Kitaev wire arises. We define the effective Hamiltonian like in Eq. (2), albeit with the exact self energy from the normal portion of the wire evaluated at zero frequency Σ_{N}(ω = 0) (see Supplementary Material). As the coupling to the reservoirs increases (the junction transparency T_{N} grows), two eigenvalues drop out from the band edge into the lower complex plane and merge at the imaginary axis at an exceptional point, much like in Fig. 2d. In contrast to the Kitaev model, however, this exceptional point is only reached if the Zeeman field exceeds a certain value , see dashed lines in 3(c,d). This B_{h} is the field required for the normal nanowire to become helical. For B > B_{h}, the normal nanowire hosts a single propagating mode, with the other spin sector completely depleted by the Zeeman field and behaves as a singlemode reservoir like the one discussed for the Kitaev wire. Note that this helical regime should be achievable using electrostatic gating, since it only requires a sufficient depletion of the nonproximitized section of the semiconductor nanowire, unscreened by the superconductor.
After crossing the exceptional point (region above thick black curves in 3[c,d]), the scattering matrix S(ω) at the trivial NS contact acquires Z = 2 purely imaginary poles. One of the two moves towards the origin. The asymptotic decay rate Γ_{0} in the limit T_{N} → 1 is not vanishing in general, so that the corresponding states should be denoted as Majorana resonances^{50}. However, in the experimentally relevant Andreev limit μ_{S} ≫ Δ, Fig. 3d, the asymptotic Γ_{0} vanishes exponentially with μ_{S}/Δ (see inset). When the wire is tuned into the regime μ_{S} ≫ Δ and the contact is made sufficiently transparent, the Majorana resonances are stabilised into proper nondecaying MDSs. For example, for the realistic parameters of the simulations in Fig. 3d the lifetime for the minimum widths (see inset) corresponds to ~0.2 microseconds. While this is already a rather long time, this is not an upper bound since even longer lifetimes can in principle be obtained by increasing μ_{S}.
The interpretation of this mechanism is the same as in the Kitaev model. While two MBSs at opposite ends of an isolated topological nanowire can be considered exact zero modes up to exponentially small corrections (in the wire length) coming from their mutual overlap, a MDSs (red in Fig. 1) also becomes an exact Majorana zero mode without any fine tuning by a suppression of its overlap with its sibling Majorana (blue in Fig. 1), which escapes into the helical reservoir. It is important to note that, in contrast to isolated topological wires, any residual overlap that remains after the exceptional point does not translate into a finite energy splitting, but rather into a residual decay rate Γ_{0}.
Physical properties of Majorana dark states
Having established the emergence of zero energy dark states at a transparent helical metaltrivial superconductor junction in the Andreev limit, we now turn to the analysis of the physical properties of said states and compare them to conventional MBSs. We will study their signatures in transport, their wavefunction locality, particlehole conjugation, their charge neutrality, uniform charge oscillations and finally, the 4π fractional Josephson effect and their nonAbelian braiding properties in SNS geometries.
Signatures in dI/dV
We start by analysing the differential conductance dI/dV through a normal tunnelling probe weakly coupled to the neighbourhood of the junction, brown in Fig. 1. In the tunneling limit, this is proportional to the local density of states at the junction at energy ε = eV, where V is the bias voltage. (Note that this is different from the differential conductance across the NS contact, which is not necessarily in the tunneling regime, see below). We computed the tunnelling dI/dV versus B and V using standard quantum transport techniques^{51} (see Supplementary Material). It is shown in Fig. 4(a–f) for several transparencies T_{N}, both far from the Andreev limit (top row) and deep into the Andreev limit (bottom row). Atop each panel, the evolution of the lowest complex eigenvalues with B is shown, with an exceptional point bifurcation at B = B_{ep}. Panel (f), with T_{N} → 1, corresponds to a cut along the top of Fig. 3(e), for which MDSs are fully developed. Their presence gives rise to sharp zero bias anomaly (ZBA) in transport at fields B > B_{h}, with a sharpness that increases exponentially with μ_{S}/Δ. This type of ZBA was the first signature of MBSs explored experimentally^{10}, though in the present context they arise far from the topological regime B ≪ B_{c}. The ZBA is not preceded by signatures of a gap closing. Note also that away from the ideal conditions T_{N} → 1, μ_{S} ≫ Δ, wide Majorana resonances are also visible in the topologically trivial regime, panels (b,c,e), albeit of finite lifetime. Injecting current through the normal reservoir [panels (g,h)] yields a very different dI/dV profile for large T_{N} (unlike for T_{N} ≪ 1), which is no longer a measurement of the local density of states. The ZBA appears in this case as a sharp dip on top of a constant 2e^{2}/h background (perfect Andreev reflection), though again only for B > B_{h}. This is consistent with general scattering theory^{30,52,53,54,55}, that predicts a dI/dV = 0 from a single mode reservoir at V → 0 if the topology is trivial. The dI/dV = 2e^{2}/h plateau is preceded by another unsplit triangular 4e^{2}/h plateau for B < B_{h}, forming a characteristic feature that should be experimentally recognizable. Note that for B > B_{c} one also obtains a sharp dip at zero, panel (g). This is due to the finite length of the superconducting wire^{29,30}, L_{S} = 1.5μm here.
Spatial localization and Majorana character
The spatial locality and Majorana selfconjugation γ = γ^{†} are assessed next, by analysing the wavefunction of the MDSs. Figure 5a shows, in red, the quasiparticle density ψ(x)^{2} = u^{2} + v^{2} (solid lines) and charge density ρ(x)^{2} = u^{2} − v^{2} (dashed lines) of the MDS marked by the white arrow in Fig. 4f (u and v are particle and hole components of its wavefunction, respectively). As discussed above, the MDS represents a nondecaying state at zero energy. The figure shows that it is furthermore well localised at the junction, decaying exponentially as ~e^{−x/ξ} with a Majorana localization length ξ = v_{F}/Δ(B)^{56} into the superconductor (see envelopes and inset in 5a). For comparison we also show in black the spatial probability ψ(x)^{2} of a conventional B > B_{c} MBS for a topological bulk at zero transparency (isolated topological wire). For both states, the charge density ρ(x) is zero, as implied by the Majorana relation γ = γ^{†}.
We now examine the charge density patterns that arise from the weak overlap of two MDSs. It was shown^{57,58} that the charge density ρ(x) = u^{2} − v^{2}, which is zero everywhere for an isolated MBS (see Fig. 5a), develops a spatial oscillatory pattern that is uniform throughout space when two MBSs approach each other in a 1D superconductor, irrespective of their particular positions. This is a very specific and nontrivial signature of MBSs that probes the state wavefunction itself and was proposed as a way to detect MBSs through charge sensing. A transparent NSN junction (with N portions coupled to reservoirs) provides a convenient geometry to study this effect in the topologically trivial regime. Two localized MDSs appear for B > B_{h} at the two ends of the S section. They weakly overlap and should thus be expected to exhibit uniform spatial charge oscillation throughout the superconductor in analogy to conventional MBSs. Figure 5(b,c) compare the charge density ρ(x) for B > B_{c} MBSs in the tunnelling limit and B_{h} < B < B_{c} in the transparent limit. Once more, we see the strong similarity between the two cases, which points to an essential equivalence between the two types of states.
Fractional Josephson effect
We next consider the fractional Josephson effect. A Josephson junction hosting a pair of conventional MBSs was shown, in the absence of quasiparticle poisoning, to develop an anomalous supercurrent I(ϕ) that is 4πperiodic in the superconductor phase difference ϕ^{6,21,22,59,60,61}. The anomalous supercurrent component is carried by the MBSs, which exhibit a parityprotected zeroenergy crossing at ϕ = π. We consider the analogous Josephson experiment in which a pair of MDSs reside at either side of a topologically trivial Josephson junction. This may be achieved in an open version of a standard Josephson junction formed in a superconducting ring, see inset of Fig. 5e. The two sides of the weak link, of transmission T_{J}, are coupled to normal reservoirs by two NS contacts that are kept open (T_{N} ≈ 1), so that each of them hosts a MDS for . A flux Φ through the superconducting loop controls the phase difference ϕ across the junction, which sustains a supercurrent I(ϕ). The main contribution to the supercurrent is carried by the ϕdependent hybridization of the two MDSs. As long as T_{J} is small enough, the MDSs will not decay, since their sibling Majoranas are essentially unperturbed by T_{J} and remain strongly delocalised into the helical reservoirs. As T_{J} is increased, however, the MDSs will become a finitelifetime resonances, as in the case of T_{N} < 1.
Figure 5d shows the computed Smatrix poles of the open Josephson junction from two hybridized decaying Majorana resonances (T_{J} is large), located at ε_{p} and . These poles represent Andreev quasibound states at the junction. Their real energy ±E_{p} is shown by the solid lines and their decay rate by the dashed lines. Each of these poles contribute with a Lorentzian to the density of states (DOS) of the junction ρ(ω). The derivative of the DOS respect to ϕ, integrated over occupied states (ω < 0) yields the supercurrent carried by these decaying states^{62}, see Supplementary Material. Around ϕ = π, we see that ±E_{p}(ϕ) crosses two new exceptional points, which in particular force ±E_{p} = 0 at ϕ = π, just like in isolated topological Josephson junctions. In practice, however, the finite decay rate Γ = Im ε_{p} of the hybridized Majorana resonances for large T_{J} (or smaller T_{N} < 1) precludes a fractional Josephson effect from developing, with 4πperiodic harmonics only expected in fast transients. This also happens in finitelength topological nanowires (see e.g. Andreev spectrum of Fig. 6a) and in Josephson junctions with quasiparticle poisoning^{63}. We emphasize moreover that the double exceptionalpoint structure around ϕ = π found here is similarly obtained in the phase dependence of a topologically nontrivial Josephson junction open to an environment^{64} and has more generally been shown to emerge from Dirac cones in photonic crystals with nonhermicity arising from radiation^{65}. As such it is already a signature in the open regime of an underlying protected crossing in the closed limit (here T_{N} → 1, T_{J} → 0).
Figure 5e shows a case closer to this limit, with T_{N} ≈ 1 and T_{J} = 0.06. The suppression of T_{J} enhances Andreev reflection from the normal leads and suppresses normal reflection processes. The residual lifetime of the hybridized MDSs is similarly suppressed (dashed line). At the same time the two exceptional points approach and merge at ϕ = π, which then appears as the protected level crossing E_{p} = 0 of a closed system. Crucially, no numberconserving perturbation can lift this crossing, just split it again into two exceptional points, always with E_{p} = 0 at ϕ = π. The two MDSs thus behave as MBSs in a closed topological Josephson junction, although their ϕ = π crossing is protected by the two underlying exceptional points^{65} and charge conjugation symmetry^{45}, instead of fermion parity (which cannot be defined in an open system). We analysed the continuous evolution of the wavefunction through the level crossing at ϕ = π for T_{J} → 0 and found that a system driven slowly through this point (but faster than the residual decay rate Γ_{0}) will evolve from its ground state to an excited state with 100% probability (green arrow in panel e). The two hybridized MDSs are indeed interchanged upon crossing ϕ = π. As a result, the Josephson current for small T_{J} becomes 4πperiodic within the residual lifetime 1/Γ_{0} of the junction MDSs. Moreover, the critical current scales as . This makes MDS supercurrent indistinguishable from that of conventional MBSs in topological Josephson junctions.
NonAbelian braiding
We finally analyse the most stringent test of the Majorana character of the MDSs, namely their nonAbelian braiding statistics. This property dictates that upon an adiabatic exchange of two spatially separated (zero energy) MBSs, the corresponding operators will undergo a transformation of the type γ_{1} → −γ_{2} and γ_{2} → γ_{1}. This transformation corresponds to a Berry phase ±π/4 for the even/odd degenerate ground states after the adiabatic exchange^{7}. If we prepare the system in its even ground state (where ‘even’ refers to the parity of the number of fermions) and perform a braiding operation, it will acquire a −π/4 global phase, . Starting from the odd ground state, which has an additional zero energy fermion , the Berry phase is the opposite . This implies that under the braiding, with ϕ_{B} = π/4 irrespective of the details of the exchange path. Together with , this implies the that γ_{1} → −γ_{2} and γ_{2} → γ_{1}. More intuitively, if instead of creating a full zeroenergy fermion on , we just create a Majorana bound state or localized at its corresponding boundary, its wavefunction will adiabatically evolve under the exchange process into the other Majorana and similarly .
In the following, we demonstrate that MDSs indeed exhibit this exact braiding statistics, just as conventional MBSs. Our aim here is to conceptually demonstrate braiding of EP Majoranas, not to discuss the optimal braiding strategy for practical quantum computing. We have thus taken the simplest possible implementation of braiding, that we explain as follows. While the spatial exchange of two Majoranas in two dimensions is conceptually clear, a subtler, yet equivalent, method is to braid them in parameter space by means of at least one additional MBS pair (ancilla)^{66,67,68}. Remarkably, this idea works even in one dimension^{68} and braiding of conventional Majoranas can be achieved by sweeping the superconducting phase difference ϕ from zero to 2π in a single Josephson junction between two finitelength topological (B > B_{c}) superconducting wires described by Eq. (3), see Fig. 6a. MBSs at the outer (noncontacted) ends of the wire will form a zero energy fermion for long enough wires (in red), while the inner MBSs (the ancilla pair, in blue) will form a fermion with ϕdependent energy. The adiabatic evolution of the even and odd ground states as ϕ is increased by 2π indeed results in a Berry phase for long L_{S}, leading to exact braiding of the outer Majoranas in this limit. The computed fidelity of the braiding process, defined as F = sin2ϕ_{B}, is shown in Fig. 6c for varying B > B_{c} and L_{S}. A typical spatial evolution under the braiding process of the two Majoranas (i.e. the adiabatically evolved starting from either or is shown in Fig. 6(e), with blue/red denoting the sign of that reveals the braiding statistics.
The equivalent braiding strategy for MDSs is implemented in a topologically trivial superconductorhelical normalsuperconductor Josephson junction^{69}. The ancilla Majoranas in this situation correspond to the two Majoranas delocalised over the helical region (blue in Fig. 6b), while the braiding Majoranas are the MDSs localised at each contact (red). Regardless of their different positions in space, the low energy Andreev spectrum in this system is essentially the same as for a fourMBS topological Josephson junction^{69} (compare Fig. 6a,b). Computing the adiabatic evolution upon increasing ϕ by 2π, we once again find that ϕ_{B} approaches π/4, i.e. exact braiding statistics like in the topological case, for good enough contact transmission T_{N}. The corresponding braiding fidelity F is shown in Fig. 6d for increasing T_{N} and B. We see that F = 1 as T_{N} → 1. Note the equivalent roles of T_{N} here and of L_{S} in the topological case: these are the relevant parameters that control the deviations of the braiding Majoranas from true zero energy bound states in each case and hence the deviations from exact braiding statistics. The spatial evolution of the braided Majoranas, shown in Fig. 6f follows a similar pattern as in the topological case, albeit for wavefunctions concentrated at the NS contacts. Hence we conclude that regarding their braiding statistics, MDSs behave once more in the same way as conventional MBSs.
Discussion
We have presented a novel approach to engineer Majorana bound states in nontopological superconducting wires. Instead of inducing a topological transition in a proximitized Rashba wire, we consider a sufficiently transparent normalsuperconductor junction created on a Rashba wire, with a topologically trivial superconducting side and a helical normal side. The strong coupling to the helical environment forces a single longlived resonance to emerge from an exceptional point at precisely zero energy above a threshold transparency. This resonance evolves as the transparency is increased further into a stable dark state localised at the junction. We have demonstrated this phenomenon both in the multimode Kitaev model and in a realistic model for a proximitized semiconducting nanowire.
The zeroenergy state emerges as the junction traverses an exceptional point at the threshold transparency and becomes robustly pinned to zero energy without fine tuning by virtue of chargeconjugation symmetry. Moreover, its residual decay rate at perfect transparency is exponentially suppressed in the experimentally relevant Andreev limit. Finally, we have shown that relevant transport and spectral properties associated to these zero energy states, here dubbed Majorana dark states, are indistinguishable from those of conventional Majorana bound states, in particular their braiding statistics.
Thus, our proposal offers a new promising strategy towards generating and detecting Majorana bound states in the lab, with potential advantages over more conventional approaches in situations where manipulating the metallic environment and contact properties proves to be simpler than engineering a topological superconducting transition. Most importantly, the condition for reaching a helical phase in the normal side, while the proximitized region of the nanowire remains in the large μ_{S} Andreev limit, is expected to be accessible experimentally. All the necessary ingredients for our proposal are already available in the lab: dramatic advances in fabrication of thin semiconducting nanowires, proximitized with conventional swave superconductors, were recently reported^{70,71}. Highly transparent, singlechannel, NS contacts and a highquality proximity effect in quantitative agreement with theory were demonstrated. Reaching the helical regime in such highquality and fully tunable devices should be within reach, so we expect that our proposal for MDSs will be soon tested.
The general connection demonstrated here between Majorana states and the bifurcation of zero energy complex eigenvalues at exceptional points in open systems offers a new perspective on the mechanisms that may give rise to Majorana states in condensed matter systems. In this sense it extends conventional strategies based on topological superconductors. It moreover expands on the extensive studies on exceptional point physics in optics, where it has been shown that state coalescence has farreaching physical consequences, such as e.g. nonAbelian geometric phases^{72,73,74}. To date, most studies of this kind have been concerned with open photonic systems under paritytime (PT) symmetry^{36,37,38} and with its spontaneous breakdown through exceptional point bifurcations. This leads to intriguing physical phenomena, such as unidirectional transmission or reflection^{38}, lossinduced transparency^{39}, lasers with reversed pump dependence and other exotic properties^{40,41,42}. Such striking optical phenomena are seemingly unrelated to the physics described in this work, but interesting connections are being made. These include open photonic systems with chargeconjugation symmetry^{45} (as opposed to PT symmetry) and which show spectral transitions analogous to the zero mode bifurcation discussed here. Also, radiationinduced nonhermicity has been demonstrated as a way to convert Dirac cones into exceptional points^{65}, a phenomenon completely analogous to the conversion of zero energy crossings of Bogoliubovde Gennes excitations in Josephson junctions into double exceptional point structures like those in Fig. 5d. Further research should extend the understanding of these interesting connections. More importantly, our study and others^{45,65} raise relevant questions, such as whether there is a deeper connection between topological transitions in closed systems and spectral bifurcations in nonhermitian systems. Answering such questions would help in advancing our understanding of the meaning of nontrivial topology in open systems.
Additional Information
How to cite this article: SanJose, P. et al. Majorana bound states from exceptional points in nontopological superconductors. Sci. Rep. 6, 21427; doi: 10.1038/srep21427 (2016).
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Acknowledgements
We are grateful to C.W.J. Beenakker, D. Pikulin, J. Sau and A. Soluyanov for illuminating discussions. We acknowledge the support of the European Research Council and the Spanish Ministry of Economy and Innovation through the JAEPredoc Program (J. C.), Grants No. FIS201123713 (P. S.J), FIS201233521 (R. A.), FIS201347328 (E.P.) and the Ramon y Cajal Program (E.P. and P.S.J).
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R.A. and P.S.J. contributed to the conceptual developments. J.C. and E.P. performed preliminary numerics and P.S.J. performed final numerics and prepared figures. All the authors discussed the results and provided scientific insight. R.A. and P.S.J. wrote the manuscript with input from the rest of authors. R.A. supervised the research.
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SanJose, P., Cayao, J., Prada, E. et al. Majorana bound states from exceptional points in nontopological superconductors. Sci Rep 6, 21427 (2016). https://doi.org/10.1038/srep21427
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