Introduction

Majorana fermions in high-energy physics are spin-1/2 particles that are symmetric with respect to charge conjugation symmetry, i.e., neutral fermions that coincide with their own anti-particles1,2,3. In condensed matter, they appear as quasiparticle excitations in superconductors, where particle-hole symmetry plays the role of charge conjugation3,4,5,6,7,8,9,10. Generally, Majorana quasiparticles are topologically protected (d − 1)-dimensional boundary excitations of a topologically nontrivial d-dimensional bulk. Specifically, 0-dimensional (0D) Majorana modes11,12,13 correspond to the end states of 1D quantum systems with proximitized superconductivity, whereas chiral and helical 1D Majorana modes correspond to the edge states of 2D unconventional superconductors or planar superconducting heterostructures14,15,16,17,18,19,20,21,22,23,24, respectively with broken or unbroken time-reversal symmetry.

Majorana quasiparticles exhibit remarkable properties such as non-abelian statistics25,26, conformal invariance7, and emergent supersymmetry (SUSY)27,28,29,30,31,32,33,34,35,36. Besides their purely theoretical appeal, Majorana quasiparticles attracted enormous interest due to their potential technological applications in quantum computing25,26,37,38,39,40,41,42,43. Quite a few experiments observed signatures compatible with the presence of spatially-separated 0D Majorana modes in nanowires44,45,46,47,48,49,50,51,52,53,54,55,56 and quantum chains of adatoms57,58,59,60,61,62,63, and chiral 1D Majorana modes in planar heterostructures24,64,65,66,67. Similar experimental signatures can be reproduced, however, by trivial Andreev bound states or by Majorana modes localized at a finite distance and with a finite overlap, known as quasi-Majorana modes68, in the presence of inhomogeneous potentials69,70,71,72,73,74.

In this work, we propose the realization of centrally extended quantum-mechanical SUSY, i.e., extended SUSY with central charges, in an experimentally-accessible condensed matter system, by employing a periodic array of partially-overlapping 0D quasi-Majorana modes to realize a Majorana chain with dispersive 1D Majorana fermions, and show how this can be achieved in proximitized semiconducting nanowires75,76,77,78,79 via periodically-modulated magnetic fields80,81,82,83,84,85 with large variations of the field intensity. We find that, for strong variations of the magnetic field amplitude, the topological mass gap \({{{{{{{\mathcal{M}}}}}}}}\) can assume alternatively positive and negative values along the wire, corresponding to topologically trivial and nontrivial segments, with partially-overlapping 0D quasi-Majorana modes localized at their boundaries and forming a 1D periodic lattice. This is the first realistic proposal to realize the Majorana chain model32,33,34,35,86 in proximitized nanowires with strong spin-orbit coupling. The system exhibits a pair of dispersive and counterpropagating 1D Majorana modes delocalized along the whole wire and separated from the higher-energy bulk states, with a mass gap that can be tuned by externally applied fields. To characterize these emergent 1D Majorana modes, we introduce the concept of pseudohelicity and unveil the existence of an extended SUSY algebra with central charges. In the massless case, we found indeed that the 1D Majorana modes are pseudohelical, i.e., have opposite Majorana pseudospin, and exhibit centrally extended SUSY, with a finite zero-energy density of states and zero-bias peak delocalized along the whole wire. We find that the signature of the emergent SUSY can be revealed by the transition from a dip G = 0 to a quantized peak G = 2e2/h in the zero-bias conductance, or by inducing an adiabatic Majorana pumping in a sliding lattice of 0D quasi-Majorana modes, with quantized transport of one quasi-Majorana mode per a half cycle. Note that zero-bias peaks are generally not expected in the presence of several 0D quasi-Majorana modes localized at a finite distance. Moreover, we identify the massless Majorana fermion with a Goldstino, i.e., the Nambu-Goldstone fermion87 associated with spontaneously broken SUSY from \({{{{{{{\mathcal{N}}}}}}}}=4\) to \({{{{{{{\mathcal{N}}}}}}}}=2\). Such a partial SUSY breaking of the extended SUSY is known to be possible only in the presence of central charges. This is perfectly compatible with our setup, in which we explicitly identify the central charges of the extended superalgebra. While the extended SUSY was proposed in several condensed matter systems88,89,90, to our knowledge, this is the first real-world realization of centrally-extended quantum-mechanical SUSY, which plays essential roles in non-perturbative aspects of quantum field theory in high energy physics91,92,93, and partial breaking of the extended SUSY algebras.

Results

Effective model: 1D lattice of 0D Majorana modes

We consider a bipartite 1D lattice of 2N 0D Majorana modes

$${{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{eff}}}}}}}}}={{{{{{{\rm{i}}}}}}}}\mathop{\sum }\limits_{j=1}^{N}\left(w{\gamma }_{Aj}{\gamma }_{Bj}+v{\gamma }_{Bj}{\gamma }_{Aj+1}\right),$$
(1)

where γAj, γBj are the Majorana operators corresponding to a single Dirac operator cj = (γBj + iγAj)/2 per unit cell, and with \(w,v\in {\mathbb{R}}\). This model is a special case of the Kitaev chain model11 if μ = 2w and t = Δ = −v. In momentum space we get

$${{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{eff}}}}}}}}}=\mathop{\sum}\limits_{k}[{c}_{k}^{{{{\dagger}}} },{c}_{-k}]\cdot {{{{{{{{\bf{H}}}}}}}}}_{{{{{{{{\rm{eff}}}}}}}}}(k)\cdot {{{{{{{\boldsymbol{\tau }}}}}}}}\cdot \left[\begin{array}{l}{c}_{k}\\ {c}_{-k}^{{{{\dagger}}} }\end{array}\right],$$
(2)

up to a constant term, with \({{{{{{{{\bf{H}}}}}}}}}_{{{{{{{{\rm{eff}}}}}}}}}(k)=\left(0,v\sin k,v\cos k-w\right)\), and τ the vector of Pauli matrices. The energy dispersion is

$${E}_{k}=| {{{{{{{{\bf{H}}}}}}}}}_{{{{{{{{\rm{eff}}}}}}}}}(k)| =\sqrt{{w}^{2}+{v}^{2}-2wv\cos k},$$
(3)

with a topological mass gap \({{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\rm{eff}}}}}}}}}=| w| -| v|\). In the continuum limit (and assuming vw > 0), the Hamiltonian coincides with a 1D Dirac equation \(H=vk\,{\tau }_{y}+\left(m{v}^{2}-\frac{v}{2}{k}^{2}\right){\tau }_{z}\) with mass gap mv2 = v − w and a quadratic correction in the momentum. The covariant form is obtained by multiplying the Hamiltonian by τz. The Dirac equation is topologically trivial or nontrivial, respectively, for mv  <  0 and mv  >  0, i.e., for \({{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\rm{eff}}}}}}}}}\, > \,0\) and \({{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\rm{eff}}}}}}}}}\, < \,0\)94.

In the massless case ∣v∣ = ∣w∣ the zero-energy eigenstates are doubly degenerate at gapless points and described by the fermionic operator \({d}_{{{{{{{{\rm{M}}}}}}}}}=({\widetilde{\gamma }}_{A}+{{{{{{{\rm{i}}}}}}}}{\widetilde{\gamma }}_{B})/2\) and its hermitian conjugate \({d}_{{{{{{{{\rm{M}}}}}}}}}^{{{{\dagger}}} }\), where the nonlocal Majorana operators are \({\widetilde{\gamma }}_{A}=(1/\sqrt{N}){\sum }_{j}{\gamma }_{Aj}\), and \({\widetilde{\gamma }}_{B}=(1/\sqrt{N}){\sum }_{j}{\gamma }_{Bj}\). The gapless state v = ±w separates two topologically inequivalent phases described by the topological invariant \({{{{{{{\rm{sgn}}}}}}}}\;{{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\rm{eff}}}}}}}}}={\pm}\!{1}\) where \({{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\rm{eff}}}}}}}}}=| w| -| v|\). The Hamiltonian also exhibits 0D Majorana end modes in the nontrivial phase ∣v∣ > ∣w∣ in the case of open boundary conditions. The two end states are \({\widetilde{\gamma }}_{{{{{{{{\rm{L}}}}}}}}}\propto {\sum }_{j}{(w/v)}^{j}{\gamma }_{Aj}\) and \({\widetilde{\gamma }}_{{{{{{{{\rm{R}}}}}}}}}\propto {\sum }_{j}{(w/v)}^{N+1-j}{\gamma }_{Bj}\) localized at the opposite ends of the chain with localization length \({\xi }_{{{{{{{{\rm{eff}}}}}}}}}=1/| \log | w/v| |\).

The massless Majorana fields (∣v∣ = ∣w∣) describe a 1D free Majorana fermion in a 1+1D conformal field theory7, which coincides with a pair of counterpropagating 1D Majorana modes. The solutions of the Dirac equation for a relativistic massless particle m = 0 form a helical pair of counterpropagating modes with opposite spin. However, our effective model is spinless: The role played by the spin degree of freedom is now played by the particle-hole degree of freedom. Hence, to characterize the properties of the 1D Majorana modes in our model, we introduce the Majorana pseudospin operator as τ/2 in analogy to the spin operator σ/2 (we use natural units). We obtain that the expectation values of the Majorana pseudospin

$$\langle {{{{{{{\boldsymbol{\tau }}}}}}}}/2\rangle =\frac{{{{{{{{{\bf{H}}}}}}}}}_{{{{{{{{\rm{eff}}}}}}}}}(k)}{2{E}_{k}},$$
(4)

have opposite directions for the two modes near the gapless point, being \(\langle {{{{{{{\boldsymbol{\tau }}}}}}}}\rangle ={{{{{{{\rm{sgn}}}}}}}}(v\sin k)\hat{{{{{{{{\bf{y}}}}}}}}}\) at k → 0, π for v = ±w, respectively. Analogously to the notion of helical modes14, i.e., a pair of counterpropagating modes having opposite spin, we introduce the notion of pseudohelical modes, as a pair of counterpropagating modes having opposite Majorana pseudospin. Hence, the two modes form a pseudohelical pair near the gapless point. In this case, elastic backscattering is suppressed since the two modes crossing at zero energy are orthogonal, analogously to the case of helical modes14. The Majorana pseudospin introduced here generalizes the Majorana polarization74,95: The expectation values of the two components 〈τx,y/2〉 coincide with the Majorana polarization (up to prefactors).

Centrally extended superalgebra and partial supersymmetry breaking

In the massless case, we find that Eq. (1) exhibits extended \({{{{{{{\mathcal{N}}}}}}}}=4\) quantum mechanical SUSY given by the combined algebra defined by the supercharges

$${{{{{{{{\mathcal{Q}}}}}}}}}_{1}=\sqrt{\frac{{{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{SUSY}}}}}}}}}}{2}}\,{d}_{{{{{{{{\rm{M}}}}}}}}}(1+P),$$
(5a)
$${{{{{{{{\mathcal{Q}}}}}}}}}_{2}=\sqrt{\frac{{{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{SUSY}}}}}}}}}}{2}}\,T(1+P),$$
(5b)

which satisfy the superalgebra \(\{{{{{{{{{\mathcal{Q}}}}}}}}}_{i},{{{{{{{{\mathcal{Q}}}}}}}}}_{j}^{{{{\dagger}}} }\}=2{\delta }_{ij}{{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{SUSY}}}}}}}}}+{{{{{{{{\mathcal{Z}}}}}}}}}_{ij}\), \(\{{{{{{{{{\mathcal{Q}}}}}}}}}_{i},{{{{{{{{\mathcal{Q}}}}}}}}}_{j}\}=\{{{{{{{{{\mathcal{Q}}}}}}}}}_{i}^{{{{\dagger}}} },{{{{{{{{\mathcal{Q}}}}}}}}}_{j}^{{{{\dagger}}} }\}=0\), \(\{P,{{{{{{{{\mathcal{Q}}}}}}}}}_{i}\}=0\), with central charges

$${{{{{{{{\mathcal{Z}}}}}}}}}_{11}=-{{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{SUSY}}}}}}}}}(1+P{(-1)}^{{d}_{{{{{{{{\rm{M}}}}}}}}}^{{{{\dagger}}} }{d}_{{{{{{{{\rm{M}}}}}}}}}}),$$
(6a)
$${{{{{{{{\mathcal{Z}}}}}}}}}_{22}=0,$$
(6b)
$${{{{{{{{\mathcal{Z}}}}}}}}}_{12}={{{{{{{{\mathcal{Z}}}}}}}}}_{21}^{{{{\dagger}}} }={{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{SUSY}}}}}}}}}\{{d}_{{{{{{{{\rm{M}}}}}}}}}(1+P),{T}^{{{{\dagger}}} }\}.$$
(6c)

Here, \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{SUSY}}}}}}}}}={{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{eff}}}}}}}}}+2h| v|\) (with h > 1) is the many-body Hamiltonian having nonnegative energy levels, \(P=\mathop{\prod }\nolimits_{j = 1}^{N}{{{{{{{\rm{i}}}}}}}}{\gamma }_{Aj}{\gamma }_{Bj}\) the fermion parity, and T the translation defined by TγAjT† = γBj, \(T{\gamma }_{Bj}{T}^{{{{\dagger}}} }={\gamma }_{Aj+1({{{{{{{\rm{mod}}}}}}}}N)}\), which satisfies {T, P} = 0 and \([T,{{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{SUSY}}}}}}}}}]=0\) for ∣v∣ = ∣w∣. (The supercharges \({{{{{{{{\mathcal{Q}}}}}}}}}_{2}\) and \({{{{{{{{\mathcal{Q}}}}}}}}}_{1}\) were introduced separately in previous works32,33, but the existence of the combined \({{{{{{{\mathcal{N}}}}}}}}=4\) superalgebra has not been previously demonstrated in Majorana chain models, up to our knowledge.) All many-body eigenstates, including the groundstate, have superpartners with opposite parity. Thus, the Witten index is zero, and SUSY is spontaneously broken27. The two degenerate groundstates are the vacuum \(\left|0\right\rangle\) and the state \(\left|1\right\rangle ={d}_{{{{{{{{\rm{M}}}}}}}}}^{{{{\dagger}}} }\left|0\right\rangle\), respectively with even and odd fermion parity. The supercharge \({{{{{{{{\mathcal{Q}}}}}}}}}_{1}\left|0\right\rangle\) annihilates both groundstates \({{{{{{{{\mathcal{Q}}}}}}}}}_{1}\left|0\right\rangle ={{{{{{{{\mathcal{Q}}}}}}}}}_{1}\left|1\right\rangle ={{{{{{{{\mathcal{Q}}}}}}}}}_{1}^{{{{\dagger}}} }\left|0\right\rangle ={{{{{{{{\mathcal{Q}}}}}}}}}_{1}^{{{{\dagger}}} }\left|1\right\rangle =0\). However, the two groundstates are superpartners with respect to \({{{{{{{{\mathcal{Q}}}}}}}}}_{2}\): Since \([T,{{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{SUSY}}}}}}}}}]=0\) and {T, P} = 0, the eigenstates \(\left|0\right\rangle\) and \(T\left|0\right\rangle\) have same energy but opposite parities, which mandates \(T\left|0\right\rangle =\left|1\right\rangle\) and \(T\left|1\right\rangle =\left|0\right\rangle\) (up to a complex phase). This yields

$${{{{{{{{\mathcal{Q}}}}}}}}}_{2}\left|0\right\rangle =\sqrt{2{{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{SUSY}}}}}}}}}}\,\left|1\right\rangle ,\;\; {{{{{{{{\mathcal{Q}}}}}}}}}_{2}^{{{{\dagger}}} }\left|1\right\rangle =\sqrt{2{{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{SUSY}}}}}}}}}}\,\left|0\right\rangle ,\;\; {{{{{{{{\mathcal{Q}}}}}}}}}_{2}^{{{{\dagger}}} }\left|0\right\rangle ={{{{{{{{\mathcal{Q}}}}}}}}}_{2}\left|1\right\rangle =0.$$
(7)

Hence, the supersymmetry \({{{{{{{{\mathcal{Q}}}}}}}}}_{1}\) is unbroken whereas \({{{{{{{{\mathcal{Q}}}}}}}}}_{2}\) is spontaneously broken: The \({{{{{{{\mathcal{N}}}}}}}}=4\) superalgebra \(({{{{{{{{\mathcal{Q}}}}}}}}}_{1},{{{{{{{{\mathcal{Q}}}}}}}}}_{2})\) is spontaneously broken down into the \({{{{{{{\mathcal{N}}}}}}}}=2\) superalgebra \({{{{{{{{\mathcal{Q}}}}}}}}}_{1}\). This mandates the presence of a Goldstino87, which we identify with the massless Majorana fermion. The zero mass gap is protected by SUSY, i.e., the gap closes if and only if the Hamiltonian exhibits SUSY. We note that the supercharge \({{{{{{{{\mathcal{Q}}}}}}}}}_{1}\) can be defined even in the presence of disorder and broken translational symmetry33,96: In this case, the corresponding Goldstino has a mass gap which closes if and only if the Hamiltonian exhibits SUSY. We note that the no-go theorem by Witten27,97 forbids partial supersymmetry breaking in extended superalgebras with zero central charges: Either all supersymmetries \({{{{{{{{\mathcal{Q}}}}}}}}}_{i}\) are broken, or they are all unbroken. However, the no-go theorem can be evaded in the presence of nonzero central charges98, which allow the partial breaking of the extended SUSY.

Proximitized semiconducting nanowire with spin-orbit coupling and periodic magnetic field

The Hamiltonian in Eq. (1) may describe the low-energy effective theory of a 1D topological superconductor with spatially-modulated fields. Specifically, we consider a semiconducting nanowire with Rashba spin-orbit coupling and coated with a conventional superconductor99, as in Fig. 1. A periodically-modulated magnetic field in the zx plane Bnm(x) is induced by an array of nanomagnets80,81,82,83,100 with magnetic moments parallel to the z-axis. In addition, we consider a uniform applied field Ba in the z-direction, which can be used to control the lengths of the nontrivial and trivial segments. The wire is described by the Hamiltonian density

$$H=\left(\frac{{p}^{2}}{2m}+\frac{\alpha }{\hslash }{\sigma }_{y}p-\mu \right){\tau }_{z}+{{{{{{{\bf{b}}}}}}}}(x)\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}+{{\Delta }}(x){\tau }_{x},$$
(8)

where σ and τ are the vectors of Pauli matrices in spin and particle-hole space, m the effective mass, α the spin-orbit coupling, μ the chemical potential, b(x) = (gμB/2)B(x) the Zeeman field in the zx plane (\({{{{{{{\bf{B}}}}}}}}(x)={{{{{{{{\bf{B}}}}}}}}}_{{{{{{{{\rm{nm}}}}}}}}}(x)+{B}_{{{{{{{{\rm{a}}}}}}}}}\hat{{{{{{{{\bf{z}}}}}}}}}\) is the total magnetic field), and Δ(x) the proximization-induced superconducting pairing. We require the wavelength λ of the periodically-modulated field to be comparable to the Majorana localization length, which is ξM ≈ (b/ESO)α/Δ and α/Δ respectively for ESO = mα2/2ℏ2 ≪ Δ and ≫ Δ (weak and strong spin-orbit coupling regimes)9,101,102. It is essential to our proposal to consider magnetic fields with large variations of the field intensity, contrarily to the well-known regime of periodically rotating fields with constant amplitude (or negligible amplitude variations), considered before80,81,103,104. Notice that periodically-modulated magnetic fields can also be induced by employing magnetic textures105,106 or domain walls107,108, while periodically-modulated chemical potentials can be obtained by a periodic modulation of the width of the superconducting coating96,109 in epitaxial 1D semiconductor-superconductor heterostructures110,111,112,113.

Fig. 1: Lattice of 0D Majorana modes in a proximitized nanowire.
figure 1

A semiconducting nanowire with Rashba spin-orbit coupling α covered by a superconducting shell, with a periodically-modulated magnetic field induced by a regular array of nanomagnets and an externally applied field Ba. The periodic modulation of the field induces topologically nontrivial (NT) segments alternating to trivial ones, which correspond respectively to negative and positive values of the local Majorana mass \({{{{{{{\mathcal{M}}}}}}}}\). 0-dimensional quasi-Majorana modes γAj and γBj localize at the boundaries between trivial and nontrivial segments, with overlaps w and v.

Full size image

If all fields are uniform, the Hamiltonian above reduces to the Oreg-Lutchyn minimal model12,13. The sign of the Majorana mass gap \({{{{{{{\mathcal{M}}}}}}}}=\sqrt{{\mu }^{2}+{{{\Delta }}}^{2}}-| b|\) characterize the trivial (\({{{{{{{\mathcal{M}}}}}}}}\, > \,0\)) and nontrivial phases (\({{{{{{{\mathcal{M}}}}}}}}\, < \,0\)) with \({{{{{{{\mathcal{M}}}}}}}}=0\) at the closing of the particle-hole gap. If the chemical potential, Zeeman field, or superconducting pairing is not uniform, one can define a local Majorana mass gap \({{{{{{{\mathcal{M}}}}}}}}(x)=\sqrt{\mu {(x)}^{2}+{{\Delta }}{(x)}^{2}}-| b(x)|\) which may be alternatively positive and negative values along the wire. In this case, segments with \({{{{{{{\mathcal{M}}}}}}}}\, > \,0\) and \({{{{{{{\mathcal{M}}}}}}}}\, < \,0\) are trivial and nontrivial with a local topological invariant \({{{{{{{\mathcal{P}}}}}}}}(x)={{{{{{{\rm{sgn}}}}}}}}\;{{{{{{{\mathcal{M}}}}}}}}(x)\). Hence, 0D quasi-Majorana modes localize at the boundaries between trivial and nontrivial segments at the nodes of the Majorana mass gap \({{{{{{{\mathcal{M}}}}}}}}(x)=0\) (see Fig. 1) with localization length ξM and mutual distance LAB and LBA. If the wavelength λ of the periodically-modulated field is comparable with the Majorana localization length ξM, the 0D quasi-Majorana mode γAj and γBj have finite overlaps \(w\propto {{{{{{{{\rm{e}}}}}}}}}^{-{L}_{AB}/{\xi }_{{{{{{{{\rm{M}}}}}}}}}}\) and \(v\propto {{{{{{{{\rm{e}}}}}}}}}^{-{L}_{BA}/{\xi }_{{{{{{{{\rm{M}}}}}}}}}}\), and realize a periodic bi-partite 1D lattice. Notice that the values of the overlaps v, w depend strongly on the distance between the nodes of the Majorana mass, i.e., the distance between neighboring 0D quasi-Majorana modes. Hence, projecting onto the subspace of Majorana operators, one obtains the effective low-energy Hamiltonian in Eq. (1), where the coupling parameters v, w coincide with the Hamiltonian matrix elements between contiguous 0D quasi-Majorana modes separated by trivial and nontrivial segments, respectively. For ∣v∣ = ∣w∣, the overlaps between contiguous 0D quasi-Majorana modes across trivial and nontrivial segments become equal, and the 1D Majorana modes become massless (gapless). We calculate the magnetic field Bnm of the nanomagnets array via the finite-element method, see Supplementary Figure 1. We use the resulting Zeeman field in all numerical calculations. For reference, we find that a reasonable approximation of the Zeeman field is

$${{{{{{{{\bf{b}}}}}}}}}_{{{{{{{{\rm{nm}}}}}}}}}(x)\approx {b}_{{{{{{{{\rm{nm}}}}}}}}}\left[\left(1-\cos (2\pi x/\lambda )\right)\hat{{{{{{{{\bf{z}}}}}}}}}-\sin (2\pi x/\lambda )\hat{{{{{{{{\bf{x}}}}}}}}}\right],$$
(9)

where bnm coincides with the magnitude of the average field along the wire, which gives \(| {{{{{{{{\bf{b}}}}}}}}}_{{{{{{{{\rm{nm}}}}}}}}}(x)| \approx | {b}_{{{{{{{{\rm{nm}}}}}}}}}| \sqrt{\left(2-2\cos (2\pi x/\lambda )\right)}\). We then discretize the Hamiltonian and take realistic values for the model parameters44,71. Moreover, we assume \({{\Delta }}(x)\propto {{{\Delta }}}_{0}\sqrt{1-B{(x)}^{2}/{B}_{{{{{{{{\rm{c}}}}}}}}}^{2}}\), where Δ0 is the superconducting gap at zero field and Bc the critical field, in order to account for the suppression of the superconducting pairing induced by the magnetic field. By numerically diagonalizing the discretized Hamiltonian, we obtain the energy En and Nambu spinor Ψn(x) of each eigenstate (see Methods section).

Figure 2(a), (b) show the local density of states (LDOS) at zero energy \(\varrho (x)=\frac{1}{\pi }{{{{{{{\rm{Im}}}}}}}}{\sum }_{n}| {{{\Psi }}}_{n}(x){| }^{2}/({E}_{n}-{{{{{{{\rm{i}}}}}}}}{{\Gamma }})\) with finite broadening Γ = 0.1Δ0 (to simulate the experimental conditions), and energy spectra as a function of the uniform applied magnetic field, in the case of periodic boundary conditions. The LDOS shows the presence of a periodic lattice of 0D quasi-Majorana modes with finite overlap, localized at the nodes of the Majorana mass gap \({{{{{{{\mathcal{M}}}}}}}}(x)=0\) [see also Fig. 2(e)]. This lattice corresponds to dispersive 1D Majorana modes with energy below the particle-hole gap and separated from the higher-energy bulk states, highlighted in Fig. 2(b), and with a Majorana mass gap equal to \({{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\rm{eff}}}}}}}}}=| w| -| v|\). When the overlaps v, w between 0D quasi-Majorana modes across the nontrivial and trivial segments become equal when ba = bSUSY (which, in first approximation, occurs when LAB ≈ LBA), the periodic lattice becomes invariant up to translations T [see Eq. (5)]. Hence, the wire exhibits SUSY and the dispersion becomes gapless (massless) with maximum LDOS at zero energy.

Fig. 2: Local density of states and energy spectra of a Majorana lattice.
figure 2

Numerically calculated local density of states (LDOS) and energy spectra as a function of the applied magnetic field intensity, showing a 1D lattice of 0D quasi-Majorana modes. a LDOS at zero energy of a nanowire in a periodically-modulated magnetic field, as a function of the position x and the external field ba applied in the z-direction, calculated with periodic boundary conditions. The peaks of the LDOS indicate the presence of 1D Majorana modes, corresponding to a periodic lattice of overlapping 0D quasi-Majorana modes localized at the boundaries between trivial (\({{{{{{{\mathcal{M}}}}}}}}\, > \,0\)) and nontrivial (\({{{{{{{\mathcal{M}}}}}}}}\, < \,0\)) segments. b Energy spectra E with dispersive 1D Majorana modes (highlighted) below the particle-hole gap. The dispersion becomes gapless (massless) when the overlaps between localized 0D quasi-Majorana modes across the nontrivial and trivial segments become equal when ba = bSUSY. c, d Same as before, but with open boundary conditions. The wire becomes nontrivial after the closing of the particle-hole gap for ba > bSUSY and exhibits 0D quasi-Majorana modes localized at its opposite ends with diverging density (out of scale). e Local Majorana mass and its nodes \({{{{{{{\mathcal{M}}}}}}}}=0\) (continuous line) as a function of the position and applied field. When ba > bNT, the Majorana mass becomes negative on the whole wire and the trivial segments disappear. Conversely, when ba < bT, the Majorana mass becomes positive on the whole wire and the nontrivial segments disappear. f Energy dispersion of the 1D Majorana modes as a function of the momentum k and the applied field. Energies and magnetic fields are in units of the superconducting gap Δ0, lengths are in units of the wavelength λ of the periodically-modulated field.

Full size image

When the applied field increases above the threshold bNT, such that \(| b(x)| \ge \sqrt{\mu {(x)}^{2}+{{\Delta }}{(x)}^{2}}\) ∀x, the Majorana mass becomes negative on the whole wire and the trivial segments disappear. Conversely, when the applied field decreases below the threshold bT, such that \(| b(x)| \le \sqrt{\mu {(x)}^{2}+{{\Delta }}{(x)}^{2}}\) ∀x, the Majorana mass becomes positive on the whole wire and the nontrivial segments disappear. In these two cases, the 0D quasi-Majorana modes at the ends of the trivial (or nontrivial) segments fuse into finite-energy Andreev-like fermionic modes. The continuous crossover between Majorana and Andreev-like modes is realized by increasing the overlaps between contiguous 0D quasi-Majorana modes at the ends of either the trivial or nontrivial segments, such that ∣v∣ ≫ ∣w∣ or ∣w∣ ≫ ∣v∣, without closing the particle-hole gap. This crossover also occurs when the wavelength λ becomes smaller than the Majorana localization length λ  ≲ ξM: This results in larger overlaps v, w between contiguous 0D quasi-Majorana modes fusing into fermionic Andreev-like modes. In the opposite regime λ ≫ ξM one has v, w → 0, which corresponds to decoupled quasi-Majorana modes with flat dispersion Ek ≈ 0.

Figure 2(c), (d) show the LDOS and the energy spectra in the case of open boundary conditions. For ba < bT and ba > bNT the local Majorana mass gap \({{{{{{{\mathcal{M}}}}}}}}(x)\) has the same sign along the wire, realizing a gapped trivial or nontrivial phase. In the latter case, 0D Majorana end modes \({\widetilde{\gamma }}_{{{{{{{{\rm{L}}}}}}}}}\), \({\widetilde{\gamma }}_{{{{{{{{\rm{R}}}}}}}}}\) localize at the opposite ends. However, for bT < ba < bNT, the local Majorana mass gap \({{{{{{{\mathcal{M}}}}}}}}(x)\) changes its sign along the wire, with 0D quasi-Majorana modes described by the effective Hamiltonian in Eq. (1). This effective Hamiltonian can be trivial or nontrivial with Majorana mass gap \({{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\rm{eff}}}}}}}}}=| w| -| v|\), as analyzed before. Hence, the nontrivial phase with 0D Majorana end modes \({\widetilde{\gamma }}_{{{{{{{{\rm{L}}}}}}}}}\), \({\widetilde{\gamma }}_{{{{{{{{\rm{R}}}}}}}}}\) is also realized for bSUSY < ba < bNT (i.e., ∣v∣ > ∣w∣). The LDOS integrated over the whole energy dispersion of the 1D Majorana mode is shown in Supplementary Figure 3. The LDOS as a function of energy below the bulk gap is shown in the Supplementary Movie.

Figure 2(e) shows the local Majorana mass \({{{{{{{\mathcal{M}}}}}}}}(x)\) as a function of the position and the applied magnetic field. For reference, we draw a continuous line at the boundary between trivial and nontrivial phases at \({{{{{{{\mathcal{M}}}}}}}}=0\), where the 0D quasi-Majorana modes localize. Figure 2(f) shows the momentum dispersion of the 1D Majorana modes calculated numerically as a function of the applied field, which corresponds to the highlighted subgap state in Fig. 2(b). The 1D Majorana modes correspond to a periodic lattice of localized and overlapping 0D quasi-Majorana modes. The mass gap of the 1D Majorana modes closes when ba = bSUSY.

Sliding lattice and Majorana pump

As demonstrated, 0D quasi-Majorana modes are pinned to the nodes of the local topological gap \({{{{{{{\mathcal{M}}}}}}}}(x)=0\) in the presence of spatially-modulated fields. This theoretically established the possibility to realize a Majorana chain model in nanowires. Moreover, this system can be employed to realize an adiabatic “Majorana pump”. On top of the field bnm(x) induced by the nanomagnets, let us apply a rotating field in the zx-plane forming an angle θ with the x-axis, and a uniform field \(-{b}_{{{{{{{{\rm{nm}}}}}}}}}\hat{{{{{{{{\bf{z}}}}}}}}}\) equal and opposite to the average field of the nanomagnets. The total field is

$${{{{{{{\bf{b}}}}}}}}(x)={{{{{{{{\bf{b}}}}}}}}}_{{{{{{{{\rm{nm}}}}}}}}}(x)-{b}_{{{{{{{{\rm{nm}}}}}}}}}\hat{{{{{{{{\bf{z}}}}}}}}}+{b}_{{{{{{{{\rm{a}}}}}}}}}[\cos \theta \,\hat{{{{{{{{\bf{x}}}}}}}}}+\sin \theta \,\hat{{{{{{{{\bf{z}}}}}}}}}].$$
(10)

If bnm(x) is approximately harmonic as in Eq. (9) one has \(| {{{{{{{\bf{b}}}}}}}}(x){| }^{2}={b}_{{{{{{{{\rm{nm}}}}}}}}}^{2}+{b}_{{{{{{{{\rm{a}}}}}}}}}^{2}-2{b}_{{{{{{{{\rm{nm}}}}}}}}}{b}_{{{{{{{{\rm{a}}}}}}}}}\sin\; (2\pi x/\lambda +\theta )\), which gives \(| b| \approx \sqrt{{b}_{{{{{{{{\rm{nm}}}}}}}}}^{2}+{b}_{{{{{{{{\rm{a}}}}}}}}}^{2}}-({b}_{{{{{{{{\rm{nm}}}}}}}}}{b}_{{{{{{{{\rm{a}}}}}}}}}/\sqrt{{b}_{{{{{{{{\rm{nm}}}}}}}}}^{2}+{b}_{{{{{{{{\rm{a}}}}}}}}}^{2}})\,\sin\; (2\pi x/\lambda +\theta )\). Thus, assuming \(\sqrt{{\mu }^{2}+{{{\Delta }}}^{2}}\approx \sqrt{{b}_{{{{{{{{\rm{nm}}}}}}}}}^{2}+{b}_{{{{{{{{\rm{a}}}}}}}}}^{2}}\) the local Majorana mass gap becomes \({{{{{{{\mathcal{M}}}}}}}}(x)=({b}_{{{{{{{{\rm{nm}}}}}}}}}{b}_{{{{{{{{\rm{a}}}}}}}}}/\sqrt{{b}_{{{{{{{{\rm{nm}}}}}}}}}^{2}+{b}_{{{{{{{{\rm{a}}}}}}}}}^{2}})\,\sin \;(2\pi x/\lambda +\theta )\) which has equally-spaced nodes at xn/λ = θ/2π + n/2. Slowly varying the applied field direction θ induces the adiabatic sliding of the 1D lattice of 0D quasi-Majorana modes, corresponding to the pumping of one 0D quasi-Majorana mode every half-turn θ → θ + π and one full fermionic state every full turn θ → θ + 2π of the applied field direction. In the case of periodic and closed boundary conditions, a half-turn of the applied field direction corresponds to the translation T of 0D quasi-Majorana modes entering the definition of the supercharges in Eq. (5). Figure 3(d) shows the intensity of the total magnetic field in Eq. (10) as a function of θ (see also Supplementary Figure 2). Figure 3(a) shows the evolution of the LDOS when the applied field direction θ turns around in the zx-plane. As the field rotates, 0D quasi-Majorana modes slide along the wire, resulting in an adiabatic pumping of 0D quasi-Majorana modes, as shown in Fig. 3(a). 0D quasi-Majorana modes translate by T at each half-turn θ → θ + π. Figure 3(b) shows the energy of the dispersive 1D Majorana modes below the gap, which corresponds to the sliding of the lattice of 0D quasi-Majorana modes. For reference, Fig. 3(c) shows the local Majorana mass \({{{{{{{\mathcal{M}}}}}}}}\) and its nodes \({{{{{{{\mathcal{M}}}}}}}}=0\) as a function of θ. This result may be generalized to the dynamical Floquet regime, with the possible realization of a finite “Majorana current” through the wire.

Fig. 3: Local density of states and energy spectra of a sliding Majorana lattice.
figure 3

Numerically calculated local density of states (LDOS) and energy spectra as a function of the applied magnetic field direction, showing a sliding 1D lattice of 0D quasi-Majorana modes. a LDOS at zero energy as a function of the position x and the magnetic field direction θ in the zx-plane, calculated with closed boundary conditions. 0D quasi-Majorana modes localize at the boundaries between trivial (\({{{{{{{\mathcal{M}}}}}}}}\, > \,0\)) and nontrivial (\({{{{{{{\mathcal{M}}}}}}}}\, < \,0\)) segments. As the field rotates, the lattice of 0D quasi-Majorana modes slides in the x-direction. b Energy spectra E with dispersive 1D Majorana modes (highlighted) below the gap. c Local Majorana mass \({{{{{{{\mathcal{M}}}}}}}}\) as a function of the applied field direction θ, with nodes at \({{{{{{{\mathcal{M}}}}}}}}=0\) (continuous lines). d Total magnetic field calculated as the superposition of the applied field Ba and the nanomagnets fields for different directions θ of the applied field (continuous lines) compared with a sliding harmonic field \(\propto \sin (2\pi x/\lambda +\theta )\) (dotted lines). Energies are in units of the superconducting gap Δ0, lengths are in units of the wavelength λ of the periodically-modulated field.

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Discussion

We theoretically proposed the realization of extended quantum-mechanical SUSY with central charges and dispersive 1D Majorana fermions in condensed matter, specifically, in a proximitized semiconducting nanowire via spatially-modulated magnetic fields. As shown in previous studies, a chain of Majorana modes exhibits both quantum mechanical SUSY32,33 and space-time SUSY34,35,86. In our work, we unveiled the presence of an additional highly-nontrivial structure, i.e., an extended quantum-mechanical SUSY with central charges. This structure emerges as the combination of two coexisting \({{{{{{{\mathcal{N}}}}}}}}=2\) superalgebras: To our knowledge, the properties of these two coexisting superalgebras have not been previously demonstrated in the Majorana chain context. Extended SUSY with central charges has been one of the most important notions in quantum field theory and string theory over the decades since the second string revolution in the 90s and the revolution of quantum field theory by Seiberg and Witten91,92,93. In spite of its great importance in formal aspects of quantum field theory and string theory, all high energy theorists regard it as a useful tool which is not directly related to reality. This is because the extended \({{{{{{{\mathcal{N}}}}}}}}=2\) SUSY does not allow chiral fermions relevant for elementary particles such as quarks and leptons. In high-energy phenomenology, only \({{{{{{{\mathcal{N}}}}}}}}=1\) SUSY and its breaking are considered. There are many proposals to realize SUSY in condensed matter, e.g., in Bose-Fermi mixtures of ultracold atoms114,115,116,117,118,119,120,121,122,123, Majorana Cooper-pair boxes124, and at the boundaries of topological superconductors or insulators30,125,126. However, our model describes the first accessible example of extended SUSY with central charges and partial spontaneous SUSY breaking realized in nature.

The action of the Goldstino is usually accompanied by higher derivative corrections determined thoroughly by a symmetry-breaking pattern, as found by Volkov and Akulov87 (see also recent works on extended SUSY127,128). Our theory should be considered as the leading order of the derivative expansion. Thus, higher derivative correction terms, if one could obtain them, should be summed up to the Volkov-Akulov type action for 1D Majorana fermions.

While the \({{{{{{{\mathcal{N}}}}}}}}=4\) SUSY conformal field theories in 1D are known to be characterized by a central charge c = 2129, the central charge in our case is c = 1, due to the presence of two helical pair of counterpropagating Majorana fermions, implying only \({{{{{{{\mathcal{N}}}}}}}}=2\) SUSY instead of \({{{{{{{\mathcal{N}}}}}}}}=4\) SUSY. This is compatible with the fact that \({{{{{{{\mathcal{N}}}}}}}}=4\) SUSY is spontaneously broken and only \({{{{{{{\mathcal{N}}}}}}}}=2\) SUSY remains, giving strong evidence of the existence of the unbroken extended \({{{{{{{\mathcal{N}}}}}}}}=2\) SUSY in our model.

We notice that electronic interactions with the external environment may contribute to the pinning of the Majorana modes to zero energy in a typical nanowire setup. Without interactions, the SUSY regime corresponds to a single point of the parameter space, which coincides with the closing of the particle-hole gap, and occurs when the externally applied magnetic field is exactly ba = bSUSY. In the presence of electronic interactions, however, the lowest energy level may become pinned to zero energy, and therefore the particle-hole gap may remain closed in an extended window of the parameter space. This may occur due to the self-interaction of the charge distribution of the Majorana modes mediated by the external environment130 in the case of a pair of Majorana modes localized at the edges of the nanowire. Perfectly spatially-separated Majorana modes are charge neutral. However, in our case, contiguous 0D quasi-Majorana modes have a finite overlap, which gives rise to a finite charge density131,132. This finite charge induces a screening charge distribution in the dielectric environment, which acts back onto the Majorana modes. This results in a self-interaction term that pushes the energy of the Majorana modes back to zero, as long as the nanowire has a larger dielectric constant of the external environment (e.g., the nanowire substrate)130. This mechanism can stabilize the SUSY by pinning the Majorana mass \({{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\rm{eff}}}}}}}}}\equiv 0\) over an extended parameter space.

To experimentally realize our proposal, the wire must be much longer than the field periodicity, which must be comparable with the Majorana localization length, i.e., L ≫ λ ≳ ξM. Moreover, variations of the gate and spin-orbit coupling fields must be negligible at length scales larger than λ, to guarantee an unbroken translational invariance at the mesoscopic level. Conversely, the physics described here is not affected by perturbations having a length scale shorter than λ (e.g., disorder). A different approach to realizing localized quasi-Majorana modes is by employing noncollinear magnetic textures or domain walls in complex magnet-superconductor heterostructures107,108. However, our proposal does not require the presence of a magnetic substrate and has the advantage of using proximitized semiconducting nanowires, which by far are the most extensively studied platform for topological superconductivity75,76,77,78,79.

In a finite wire with 2N 0D Majorana modes with open boundary conditions and at zero temperature, the differential conductance exhibits 2N − 1 zeros and 2N quantized peaks G = 2e2/h, and the zero-bias conductance is zero133. In an infinite wire with identical couplings (v = w) instead, the conductance shows a zero-bias peak G = 2e2/h133, which corresponds to the tunneling into the delocalized Majorana mode at zero energy. For v ≠ w, the energy of the Majorana modes is lifted by the broken SUSY, and therefore there is no Majorana mode available at zero energy: In this case, the zero-bias conductance is zero. Hence, for a sufficiently long Majorana chain or equivalently in the case of closed boundary conditions (i.e., in a loop geometry), the transition between a zero-bias dip G = 0 to a peak G = 2e2/h signals the onset of SUSY at v = w. This transition should be observable in sufficiently long nanowires or in a setup with loop geometry (i.e., with closed boundary conditions) by varying the applied magnetic field close to the SUSY point ba = bSUSY. However, these signatures may be difficult to distinguish from conductance peaks induced by disorder, impurities, or finite-size effects. The characterization of the signatures of SUSY in the conductance will be the subject of future work. Stronger experimental signatures are the presence of spatially-periodic peaks in the LDOS along the whole wire, and their adiabatic evolution in the Majorana pump regime, obtained by varying the applied magnetic field direction. These signatures can be obtained by locally probing the differential conductivity in multiterminal setups56,134,135,136, or by scanning tunneling microscopy (STM) in epitaxial 1D heterostructures110,111,112,113. Other nonlocal fingerprints of SUSY can be provided by the signatures of the closing of the bulk particle-hole gap with a finite density of states on the whole wire. In particular, these signatures can be revealed via a tunneling probe placed at the bulk of the nanowire56, by the quantized peak of the thermal conductance and electrical shot noise at the transition, and the doubling of the magnetoconductance oscillations in an Aharonov-Bohm ring geometry137, or by the peak of the 4π component of the Josephson current in a superconducting ring geometry138.

We also mention that the proposed experimental protocol to realize an adiabatic Majorana pumping in a sliding lattice of 0D quasi-Majorana modes may suggest new methods of braiding Majorana modes in 1D nanowire networks, which is the next milestone in the route to fault-tolerant topological quantum computation25,26,37,38,39,42,43.

Concluding, we proposed an experimentally accessible realization of a Majorana chain with emergent SUSY and with dispersive 1D Majorana fermions in proximitixed nanowires via spatially-modulated magnetic fields. In this system, we demonstrated the presence of an extended \({{{{{{{\mathcal{N}}}}}}}}=4\) SUSY with central charges and we identified the massless 1D Majorana fermions as the Nambu-Goldstone fermions (Goldstinos) associated with the spontaneous partial breaking of SUSY. Their experimental signatures are the finite LDOS at zero energy (zero-bias peak) delocalized on the whole length of the wire, and the dip-to-peak transition in the zero-bias conductance. This has to be contrasted with zero-bias peaks of 0D Majorana end modes, localized only at the ends of the wire, and to the general case of 0D quasi-Majorana modes, whose energy is lifted by their finite overlap. We finally showed how to realize an adiabatic Majorana pump by varying the applied magnetic field direction, which induces a sliding lattice of 0D quasi-Majorana modes with quantized transport of one quasi-Majorana mode per a half cycle. The manipulation of Majorana modes via spatially-modulated fields may lead to the realization of alternative non-abelian braiding protocols.

Methods

The numerical results were obtained by discretizing Eq. (8) into a tight-binding model and diagonalizing the resulting Hamiltonian. The LDOS in the main article and in the Supplementary Information was calculated from the spectra as a function of energy. The momentum dispersion was obtained by Fourier-transform the tight-binding Hamiltonian and diagonalizing the resulting Hamiltonian in the momentum basis. In agreement with previous works44,71, we consider an InSb nanowire proximitized by a conventional superconductor, and take m = 0.015 me, α = 1 eV Å, b/B = 1.5meV T−1 (g ≈ 50), Bc = 3T, and Δ0 = 1 meV at zero magnetic field. In this regime, the Majorana localization length is estimated to be ξM ≈ 200–300 nm depending on the applied field ba. The magnetic field induced by the array of nanomagnets was calculated by numerical integration over a finite mesh. We considered a periodic array of 250 nm × 250 nm × 250 nm nanomagnets placed at a mutual distance 750 nm and at d = 250 nm from the wire, with remnant magnetic field BR ≈ 1 T and parallel magnetic moments in the z direction. See Supplementary Note 1 and Supplementary Figs. 1 and 2 for more details.