Introduction

Majorana modes (MMs) localized at the vortex cores of two-dimensional (2D) or at the ends of one-dimensional (1D) topological superconductors (TSs)1,2,3,4,5,6,7 are potential building blocks for topological quantum computing8,9,10,11 and other exotic quantum systems that effectively simulate high-energy theories such as supersymmetry (SUSY)12,13,14,15,16,17,18,19 and synthetic horizons in the Sachdev-Ye-Kitaev (SYK) model20,21,22 (see also refs. 23,24,25,26,27,28,29,30,31,32,33,34,35,36). Specifically, 2D TSs1 realized with topological insulator (TI) or quantum spin Hall insulator/superconductor heterostructures3, transition metal dichalcogenides37,38, iron pnictides39,40,41 or other oxypnictide superconductors42, or magnet-superconductor hybrid systems43,44,45,46 may provide a flexible platform to exploit the nonabelian exchange statistics of MMs8, due to the ability to manipulate vortex cores in a 2D space, and detect them through scanning tunneling microscopy. Conversely, 1D TSs2, e.g., proximitized quantum wires6,7,47,48,49 arranged in 2D networks, offer a simplified but less flexible setup to perform braiding50,51,52.

To combine the flexibility of braiding in 2D with the conceptual simplicity of 1D platforms and overcome their limitations, here we introduce topologically nontrivial stripes (TNSs) induced by inhomogeneous superconducting states53,54,55 where the gauge-invariant phase rotates in a regular pattern. The rotating phase effectively generates quasi-1D structures within the 2D system, where the topological invariant assumes alternatively trivial and nontrivial values as a function of the phase. This results into a striped 2D TS with emergent TNSs equivalent to 1D TSs, localized at 1D lines where the order parameter phase is homogeneous and the topological invariant is nontrivial, so that a quasi-1D topological superconducting state emerges. Here, pointlike (0D) MMs localize at the ends of the TNSs, whose distance and direction can be manipulated by varying the in-plane field magnitude and direction. These highly nonlocal MMs offer multiple and flexible ways to implement braiding due to the possibility of moving and rotating the stripes in a 2D space.

Specifically, we consider a TI film in a magnetic field56,57,58 where the surface states are gapped out by proximity with a conventional superconductor, allowing the realization of second-order TSs59 with Majorana hinge modes. We show that these (1D) Majorana hinge modes transmute into pointlike (0D) Majorana corner modes localized at the end of TNSs induced by an inhomogeneous superconducting order. Hence, we describe braiding and fusion protocols implemented by joining, splitting, and moving stripes via external gates or magnetic force microscopy, and rotating them by rotating the magnetic field. Furthermore, we show how configurations with several stripes induce a regularly-spaced array of MMs, realizing emergent quantum mechanical SUSY, zero-energy multi-locational MMs60 delocalized on multiple spatially separated points, Yang-Lee anyons61 with non-unitary and nonabelian statistics, and the SYK model22 reproducing the maximally-chaotic dynamics of black holes.

Results

Creation of topologically nontrivial stripes

The boundary mean-field Hamiltonian describing the proximitized surface states at the top and bottom of a TI in a magnetic field as sketched in Fig. 1a is56,57,58,62

$$\begin{array}{ll}{\mathcal{H}}=\int\,{\rm{d}}{\bf{r}}\,{\psi }^{\dagger }({\bf{r}})H\psi ({\bf{r}})-\sum\limits_{\tau }\left({\Delta }_{\tau }({\bf{r}}){\psi }_{\tau \downarrow }({\bf{r}}){\psi }_{\tau \uparrow }({\bf{r}})+{\rm{h.c.}}\right)\\\qquad\qquad\qquad\qquad\qquad +\frac{1}{U}\int\,{\rm{d}}{\bf{r}}\,\sum\limits_{\tau }| {\Delta }_{\tau }({\bf{r}}){| }^{2},\end{array}$$
(1)

where \(\psi ({\bf{r}})={\left[{\psi }_{1,\uparrow }({\bf{r}}),{\psi }_{1,\downarrow }({\bf{r}}),{\psi }_{2,\uparrow }({\bf{r}}),{\psi }_{2,\downarrow }({\bf{r}})\right]}^{\intercal}\) are the electron field operators with ↑↓ indexes for spin, τ = 1, 2 for pseudospin (i.e., top and bottom surfaces) and

$$H=(m^{\prime} {{\bf{p}}}^{2}+m){\tau }_{x}-v{\bf{p}}\times {\boldsymbol{\sigma }}{\tau }_{z}-\mu +{\bf{b}}\cdot {\boldsymbol{\sigma }},$$
(2)

where m and \(m^{\prime}\) are the tunneling parameters depending on the layer thickness56,63, v the Dirac cone velocity, p the momentum on xy plane, b the Zeeman field, μ the chemical potential, and σi, τi the Pauli matrices in spin and pseudospin space. Here, Δτ(r) is the mean-field order parameter of the surface states, i.e., at the interface between the superconductor and the TI, and which can be calculated self-consistently58,64,65 as \({\Delta }_{\tau }({\bf{r}})=-U\left\langle 0\right\vert {\psi }_{\tau \downarrow }({\bf{r}}){\psi }_{\tau \uparrow }({\bf{r}})\left\vert 0\right\rangle\),where U the superconducting pairing strength describing the on-site electron-electron attractive interaction within the surface states. In the normal regime with unbroken time-reversal symmetry (i.e., Δ = b = 0), the Hamiltonian is unitarily equivalent56,63 to the Bernevig, Hughes, and Zhang model for a 2D quantum spin Hall TI66 with energy dispersion having two spin-degenerate branches with gap 2∣m∣ at zero momentum and a topologically nontrivial phase for \(mm^{\prime} \,<\, 0\). The TI surface states exhibit a gap 2(b − ∣m∣) at zero momentum for b ≠ ∣m∣ and remain gapped at finite momenta for finite out-of-plane fields bz > 0.

Fig. 1: Topological insulator film with top and bottom surfaces proximitized by conventional superconductors.
figure 1

a Topological insulator film sandwiched between two conventional superconductors. b Majorana hinge modes in the nontrivial phase at finite field and uniform pairing (we removed the superconductors for clarity). c Topologically nontrivial stripes and Majorana corner modes in the nonuniform superconducting phase. The LO order parameter is modulated as Δ(r) ∝ eiQ⋅r. The Cooper pairs momentum Q is perpendicular to the in-plane magnetic field bxy.

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Superconductivity emerges when the Cooper instability opens a gap at the Fermi level lying within the conduction band, assuming ∣μ∣ > b − ∣m∣. The symmetry between the top and bottom surfaces mandates ∣Δ1(r)∣ = ∣Δ2(r)∣. Hence, by choosing a gauge where the phases of the order parameter on the top and bottom surfaces are opposite, one can write Δ1(r) = Δ2(r)* up to a gauge transformation. We can thus write Δ1(r) = Δ(r) = ∣Δ(r)∣eiϕ(r) and Δ2(r) = Δ(r)* = ∣Δ(r)∣e−iϕ(r), where 2ϕ(r) is the gauge-invariant phase difference between top and bottom surfaces. The mean-field Bogoliubov-de Gennes Hamiltonian is \({{\mathcal{H}}}_{{\rm{BdG}}}=\frac{1}{2}\int{\rm{d}}{\bf{r}}\,{{{\Psi }}}^{\dagger }({\bf{r}})\cdot {H}_{{\rm{BdG}}}\cdot {{\Psi }}({\bf{r}})\) with

$$\begin{array}{l}{H}_{{\rm{BdG}}}\,=\,\left[(m^{\prime} {{\bf{p}}}^{2}+m){\tau }_{x}-v{\bf{p}}\times {\boldsymbol{\sigma }}{\tau }_{z}-\mu \right]{\upsilon }_{z}+{\bf{b}}\cdot {\boldsymbol{\sigma }}\\ \qquad\quad\,+\,\vert \Delta ({\bf{r}})\vert \left[\cos (\phi ({\bf{r}}))\,{\upsilon }_{x}+\sin (\phi ({\bf{r}}))\,{\tau }_{z}{\upsilon }_{y}\right],\end{array}$$
(3)

where \({{\Psi }}({\bf{r}})={\left[\psi ({\bf{r}}),{\sigma }_{y}{\psi }^{\dagger }({\bf{r}}){\sigma }_{y}\right]}^{\intercal}\), and υi the Pauli matrices in particle-hole space.

Let us first assume uniform superconducting pairing Δ(r) = Δeiϕ with Δ > 0. Topologically nontrivial phases with particle-hole symmetry and broken time-reversal symmetry (class D) in 2D are labeled by the Chern number of the quasiparticle excitation gap \(c\in {\mathbb{Z}}\). The gap closes when \(| {m}^{2}+{\mu }^{2}+{\Delta }^{2}-{b}^{2}| =2| m| \sqrt{{\mu }^{2}+{\Delta }^{2}{\sin }^{2}\phi }\), and remains open at finite momenta for bz ≠ 0 and ϕ ≠ 0. For ϕ = π/2, the quasiparticle excitation gap \(2\min (| | m| -| b\pm \sqrt{{\mu }^{2}+{\Delta }^{2}}| | )\) closes at zero momentum with a quantum phase transition each time that any of the quantities \(b\pm m\pm \sqrt{{\mu }^{2}+{\Delta }^{2}}\) change sign. This condition divides the parameter space into topologically distinct phases separated by the closing of the quasiparticle excitation gap, where we calculate the Chern number numerically67: We thus found a trivial phase at weak fields, where \(| m| > b+\sqrt{{\mu }^{2}+{\Delta }^{2}}\) or \(\sqrt{{\mu }^{2}+{\Delta }^{2}} > b+| m|\), a nontrivial phase with ∣c∣ = 2 at strong field \(b > | m| +\sqrt{{\mu }^{2}+{\Delta }^{2}}\), and a nontrivial intermediate phase with ∣c∣ = 1 where no energy scale dominates, i.e., when \(| m| +\sqrt{{\mu }^{2}+{\Delta }^{2}} > b > | | m| -\sqrt{{\mu }^{2}+{\Delta }^{2}}|\), or equivalently \(b+| m| > \sqrt{{\mu }^{2}+{\Delta }^{2}} > | b-| m| |\), or \(b+\sqrt{{\mu }^{2}+{\Delta }^{2}} > | m| > | b-\sqrt{{\mu }^{2}+{\Delta }^{2}}|\). The nontrivial phases persist for ϕ ≠ π/2 as long as the quasiparticle excitation gap remains open (see also Supplementary Note 2). The parity of the topological invariant68,69\(\nu =c\,\mathrm{mod}\,\,2\) is given by \({(-1)}^{\nu }={\prod }_{{\bf{k}}}\,{\rm{sgn}}\,\left({\rm{pf}}\,\left({H}_{{\rm{BdG}}}({\bf{k}}){\sigma }_{y}{\upsilon }_{y}\right)\right)\) where HBdG(k) is the Hamiltonian density as a function of the momentum eigenvalues k with the product spanning over the time-reversal symmetry points of the Brillouin zone, giving

$${(-1)}^{\nu }={\rm{sgn}}\left(\vert {m}^{2}+{\mu }^{2}+{\Delta }^{2}-{b}^{2}\vert -2| m| \sqrt{{\mu }^{2}+{\Delta }^{2}{\sin }^{2}\phi }\right).$$
(4)

Since the effective boundary Hamiltonian in Eq. (2) describes the surface states of a 3D TI, these nontrivial gapped phases are 3D second-order TS with Majorana hinge modes, i.e., gapless modes on the hinges59, as in Fig. 1b.

For zero magnetic fields or fields parallel to the z-axis, the SO(2) rotational symmetry in the xy plane is unbroken: this allows the creation of Cooper pairs with zero momentum Q = 0 formed by electrons with opposite spin and opposite momenta. However, in the presence of a finite spin-orbit coupling term ∝ p × σ, a finite in-plane magnetic field b ⋅ σ (Zeeman term) shift electrons with opposite spin in opposite directions k → k ± Q/2, with the momentum Q perpendicular to the in-plane field and Q ≈ 2bxy/v at large fields. In the Pauli limit, neglecting the orbital pair-breaking mechanism, this allows the creation of Cooper pairs with finite momentum Q, formed by electrons with opposite spin and momentum eigenvalues k and − k + Q, described by a nonuniform order parameter Δ(r) that depends periodically in space with a wavelength λ = 2π/Q. The simplest spatial dependence compatible with the symmetries of the system considered here58 is

$$\Delta ({\bf{r}})={\Delta }_{0}\left[\cos \theta \cos ({\bf{Q}}\cdot {\bf{r}})+{\rm{i}}\sin \theta \sin ({\bf{Q}}\cdot {\bf{r}})\right],$$
(5)

with Δ0 > 0 and 0 ≤ θ ≤ π/2 (up to a gauge transformation) determined by the minimum of the free energy \({\mathcal{F}}=\langle {\mathcal{H}}\rangle\) at zero temperature. The order parameter has a total magnitude \(| \Delta ({\bf{r}})| ={\Delta }_{0}\sqrt{(1+\cos (2\theta )\cos (2{\bf{Q}}\cdot {\bf{r}}))/2}\), having minima and maxima for any θ ≠ π/4 along the 1D planes parallel to the in-plane field, which we call respectively nodal and antinodal lines, satisfying Q ⋅ r = nπ/2 for \(n\in {\mathbb{Z}}\). Its phase \(\phi ({\bf{r}})=\arg \Delta ({\bf{r}})\) is spatially modulated if θ ≠ 0, π/2, being \(\tan (\phi ({\bf{r}}))=\tan \theta \tan ({\bf{Q}}\cdot {\bf{r}})\) giving \({\sin }^{2}(\phi ({\bf{r}}))=0,1\) for Q ⋅ r = nπ/2. One can verify that Δ(r, − θ) = Δ(r, θ)*, Δ(r, π/2 − θ) = − Δ(r, π/2+θ)*, and that \(\Delta ({\bf{r}},\pi /4-\theta )={\rm{i}}\Delta {({\bf{r}}{^\prime} ,\pi /4+\theta )}^{* }\) with \({\bf{r}}{^\prime} =\pi {\bf{Q}}/2{Q}^{2}-{\bf{r}}\). Consequently, H(α + θ) and H(α − θ) are unitarily equivalent and thus have the same energy spectra, which mandates \({\mathcal{F}}(\alpha +\theta )={\mathcal{F}}(\alpha -\theta )\) for α = 0, π/4, π/2. This mandates the presence of stationary points \(\delta {\mathcal{F}}(\theta )=0\) for θ = 0, π/2, and π/4 (see also Supplementary Note 3). The cases θ = 0, π/2 correspond to Larkin-Ovchinnikov (LO) orders with a constant phase ϕ(r) = 0, π/2 and magnitude \({\Delta }_{0}| \cos ({\bf{Q}}\cdot {\bf{r}})|\) and \({\Delta }_{0}| \sin ({\bf{Q}}\cdot {\bf{r}})|\), respectively, which becomes zero at the nodal lines and reaches its maximum Δ0 at the antinodal lines. The case θ = π/4 instead corresponds to a Fulde-Ferrel (FF) order with a constant magnitude \({\Delta }_{0}/\sqrt{2}\) and a phase ϕ(r) = Q ⋅ r giving \({\sin }^{2}(\phi ({\bf{r}}))=0,1\) respectively for Q ⋅ r = nπ and Q ⋅ r = π/2 + nπ. The 1D lines defined by constant Q ⋅ r have constant order parameter Δ = Δ(r) and are described by an effective 1D Hamiltonian \({H}_{{\rm{1D}}}(r)=\left[-v{p}_{x}{\sigma }_{y}{\tau }_{z}+(m{\prime} {p}_{x}^{2}+m){\tau }_{x}\right]{\upsilon }_{z}+{\bf{b}}\cdot {\boldsymbol{\sigma }}-{\Delta }_{0}\,{\tau }_{z}{\upsilon }_{y}| \Delta | (\cos \phi \,{\upsilon }_{x}+\sin \phi \,{\tau }_{z}{\upsilon }_{y})\), for in-plane fields in the x direction, which is equivalent to Eq. (3) when one takes py = 0. In symmetry class D in 1D, topologically inequivalent phases are labeled by a \(\nu \in {{\mathbb{Z}}}_{2}\) topological invariant. By dimensional reduction, ν must coincide with the parity of the topological invariant in 2D defined in Eq. (4): Hence, there is only one nontrivial phase in 1D, realized when ν = 1 in Eq. (4), as long as the quasiparticle excitation gap remains open at all momenta (see Supplementary Fig. 3).

TNSs are realized when 1D lines Q ⋅ r = nπ/2 become topologically inequivalent. For θ = 0, π/2, the order parameter phase is constant ϕ = 0, π/2: for θ = ϕ = 0, the gap closes at finite momenta in the regime where ν = 1, preventing the realization of a nontrivial gapped phase; for θ = ϕ = π/2 instead, the gap is always open for bz ≠ 0, and TNSs may emerge when nontrivial phases are realized on the antinodal lines Δ(r) = Δ0 for \(b+| m| > \sqrt{{\Delta }_{0}^{2}+{\mu }^{2}} > | b-| m| |\), and trivial phases on the nodal lines Δ(r) = 0 for ∣b − ∣m∣∣ > ∣μ∣ (nodal lines are Δ(r) = 0 and thus cannot realize a nontrivial TS). However, this phase is not physical since the superconductivity can only be realized when the Fermi level lies within the conduction band, i.e., for ∣μ∣ > b − ∣m∣. For θ ≈ π/4 instead, the order parameter is \(\Delta ({\bf{r}})\,\approx\, {\Delta }_{0}/\sqrt{2}\) which corresponds to trivial and nontrivial phases with \(\phi ({\bf{r}})={\bf{Q}}\cdot {\bf{r}}=0,\pi /2\,\mathrm{mod}\,\,\pi\), respectively, provided that

$$2| m| \sqrt{{\mu }^{2}+{\Delta }_{0}^{2}/2} \,>\, | {m}^{2}+{\mu }^{2}+{\Delta }_{0}^{2}/2-{b}^{2}| > 2| m| | \mu | ,$$
(6)

as it follows from Eq. (4). The resulting TNSs are quasi-1D nontrivial regions close to the 1D lines Q ⋅ r = πn parallel to the in-plane field and effectively equivalent to 1D TSs. If stripes extend along the whole surface, reaching the hinges, there will be a MM at each end of the stripe, as in Fig. 1c. These end modes can also be seen as the corner modes of the effectively 2D TSs obtained by extending the 1D lines along the z direction, resulting in 2D planes parallel to the in-plane field and the z-axis, cutting the 3D TI into 2D slices. Hence, the confinement of the 2D boundary Hamiltonian into a 1D Hamiltonian describing the TNSs corresponds to the confinement of the surface states of a 3D second-order topological phase (with hinge modes) into the edge states of a 2D second-order topological phase with corner modes defined by the planes with Q ⋅ r = πn.

The formation of quasi-1D topological superconducting stripes and pointlike MMs at their ends is a consequence of dimensional reduction70. The quasi-1D stipes are indeed narrow 2D regions which are topologically nontrivial, with a length determined by the system size (or by the presence of domain walls) and a width d < λ/2 coinciding with the width of the region where the phase of the order parameter ϕ is such that ν = 1 in Eq. (4). If their width is comparable with their length, these 2D nontrivial regions will exhibit 1D edge modes at their border on all four sides; however, when their width becomes narrow enough, the edge states along two opposite sides will come closer and begin to overlap in space, opening a finite energy gap as a result of their finite overlap. In particular, if the width is smaller or comparable to the Majorana localization length d ≲ ξ, only a single quantization channel will become available. In this regime, only a single pointlike mode may exist at each end of the stripe. The dimensional reduction from a 2D to a 1D topological state requires stripes with a width smaller than their length and smaller than the Majorana localization length so that only one single 1D channel is present. On the other hand, their spatial separation, given by the distance between neighboring stripes, must be larger than or comparable to the Majorana localization length λ/2 ≳ ξ so that MMs remain spatially separated. Generally, one has ξ ~ b/Δ for 1D TSs71,72.

As explained, symmetry arguments alone restrict the possible states to θ = 0, π/2 (LO states) and θ = π/4 (FF state), but only the FF state with θ = π/4 can exhibit TNSs. We find numerical evidence that the state that fulfills the self-consistence equation at zero temperature has an order parameter which is approximately equal to Eq. (5) with θ = π/4. Indeed, we calculate the order parameter self-consistently at zero temperature and as a function of the spatial coordinate as \({\Delta }_{\tau }({\bf{r}})=-U\left\langle 0\right\vert {\psi }_{\tau \downarrow }({\bf{r}}){\psi }_{\tau \uparrow }({\bf{r}})\left\vert 0\right\rangle\) using Eq. (5) with several choices of θ and with Q = 2bxy/v as the initial guess of the self-consistent calculation for realistic choices of the system parameters for Bi2Te373, proximitized with NbTiN or NbSe2, compatible with Eq. (6). The resulting order parameter obtained self-consistently at zero temperature is approximately equal to Eq. (5) with θ = π/4, corresponding to an FF order with almost constant magnitude and nonuniform phase, excluding regions close to the boundaries of the system, where the magnitude of the order parameter is slightly suppressed. This result is in agreement with the results of ref. 58, which found that FFLO states with θ ≈ π/4 are stable also at finite temperature and for large in-plane magnetic fields bxy (The ansatz for the order parameter in Eq. (5), which describes a generic FFLO state interpolating between an FF state (for θ = 0, π/2) and an LO state (for θ = π/4) coincides with the ansatz in ref. 58, where the order parameter is parameterized in terms of \(b=\cos (\theta )\). In ref. 58, it is found b = 0.77, which is approximately equal to \(b=\cos (\theta )=\cos (\pi /4)=1/\sqrt{2}\approx 0.707\), indicating an LO state although, in that paper, the state is always called an FF state for any choice of b). Therefore, the superconducting order self-tunes to support the TNS phase, which is therefore a self-organized topological state, in this regard analogous to magnetic adatom chains with a spin helical order self-tuned to support the topological phase74. Figure 2a shows the local density of states (LDOS) at zero energy in the TNSs regime calculated numerically. The peaks in the LDOS indicate MMs localized at the ends of the 1D TNSs at ϕ(r) = ±π/2.

Fig. 2: Local density of states (LDOS) at zero energy calculated numerically for a proximitized topological insulator film in the inhomogeneous superconducting phase in a system of size 162 × 54 lattice sites with open boundary conditions.
figure 2

a Topologically nontrivial stripes corresponding to the order parameter phase ϕ(r) = ± π/2 with Majorana modes at their ends separated by a distance λ/2 = π/Q ≈ πv/2bxy. b A single stripe obtained by tuning the in-plane field such that λ is approximately equal to the width of the system. c same as (b), but with a trivial region in the middle, acting as a domain wall splitting the stripe into two. d same as (c), but with an additional trivial region on the left, acting as a pointlike defect further splitting the left stripe into two. e and f same as (c) and (d), but rotating the in-plane field on the left, (e) rotating the stripe, and (f) rotating two stripes around each other. Arrows indicate the direction of the Cooper pair momentum Q and of the in-plane field bxy. Areas enclosed by dotted lines indicate regions with μ = 0 suppressing the superconducting order. The order parameter, calculated self-consistently, is compatible with θ = π/4 in Eq. (5).

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Manipulation of topologically nontrivial stripes

TNSs can be manipulated in several ways. Rotating the magnetic field around the z-axis (perpendicular to the surface) changes the in-plane field direction and hence the stripes direction, while rotating the field in the xy-plane changes the in-plane field magnitude and hence the distance λ/2 between the stripes. Moreover, topologically trivial regions ν = 0 can behave as domain walls or pointlike defects that confine the stripes or split a single stripe into two. Trivial regions can be created by locally increasing the tunneling m between the two TI surfaces (by locally modifying the TI layer thickness), such that ∣m∣ ≫ ∣μ∣, ∣Δ∣, ∣b∣ or, alternatively, decreasing the chemical potential ∣μ∣ < b − ∣m∣ by using external gates, driving the Fermi level out of the conduction band and thus suppressing the superconducting order, so that the first term in Eq. (4) dominates. Trivial regions can also be created by suppressing the magnetic field since Eq. (4) yields ν = 0 for b = 0. Furthermore, isolated stripes are obtained by tuning the distance λ/2 such that only a single stripe fits within the TNS phase (confined by the system edges or by domain walls), as in Fig. 2b.

Multilocational modes, SUSY, SYK model, and Yang-Lee anyons

MMs at the ends of TNSs hybridize within a low-energy manifold of dimension 2N, forming highly nonlocal modes delocalized over spatially separated points described by the effective Hamiltonian

$${{\mathcal{H}}}_{{\rm{eff}}}={\rm{i}}w\mathop{\sum }\limits_{n=1}^{N-1}\sum\limits_{s=L,R}{\gamma }_{s,n}{\gamma }_{s,n+1}+{\rm{i}}w^{\prime} \mathop{\sum }\limits_{n=1}^{N}{\gamma }_{L,n}{\gamma }_{R,n},$$
(7)

with γL,n and γR,n the modes on the left and right ends, w ∝ e−λ/2ξ and \(w{\prime} \propto {{\rm{e}}}^{-l/\xi }\) the couplings between modes on the same side (at a distance λ/2) and on opposite sides of the stripes (at a distance l), respectively, with \(w,w^{\prime} > 0\) up to a gauge transformation. This manifold can exhibit nonlocal fermionic modes at exactly zero energy, even at finite sizes. For N → ∞ (or equivalently, in a setup with periodic boundaries), the MMs realize two translationally invariant lattices which are decoupled for \(w{\prime} =0\). In this case, the N MMs of each lattice are degenerate under translations and hybridize into two MMs \({\widetilde{\gamma }}_{1,2}\) (forming a single fermionic mode) at zero energy, delocalized into N spatially separated points corresponding to the ends of the TNSs. This results in quantum mechanical SUSY14 or space-time SUSY in the presence of many-body interactions12,13,16. For \(w^{\prime} > 0\), the two fermionic modes (one for each side) hybridize at finite energy. The fractionalization of the fermionic degrees of freedom and the emergence of quantum-mechanical SUSY is also observed in finite systems N < ∞ with open boundary conditions. Indeed, we find that if \(w=w{^\prime}\) and N = 3m + 2 = 2, 5, 8, …, the two nonlocal modes

$${\widetilde{\gamma }}_{1}=\frac{1}{\sqrt{2m+2}}\mathop{\sum }\limits_{n = 0}^{m}{\kappa }_{n}({\gamma }_{L,3n+1}+{\gamma }_{R,3n+2})+{\kappa }_{n-1}({\gamma }_{R,3n+1}-{\gamma }_{L,3n+2}),$$
(8)
$${\widetilde{\gamma }}_{2}=\frac{1}{\sqrt{2m+2}}\mathop{\sum }\limits_{n = 0}^{m}{\kappa }_{n}({\gamma }_{R,3n+1}-{\gamma }_{L,3n+2})+{\kappa }_{n+1}({\gamma }_{L,3n+1}+{\gamma }_{R,3n+2}),$$
(9)

with \({\kappa }_{n}=\cos (n\pi /2)\), have exactly zero energy, each delocalized into 2(m + 1) = 2(N + 1)/3 spatially separated points. This regime \(w{^\prime} =w\) is realized when the length of the stripes is comparable with the distance between mutual stripes (see Supplementary Note 6). This setup can be advantageous when the system dimensions cannot be stretched indefinitely. Also, we find that if \(w^{\prime} \to 0\) and N = 2m + 1, the two nonlocal modes

$$\begin{array}{ll}{\widetilde{\gamma }}_{1}=\frac{1}{\sqrt{m+1}}\mathop{\sum }\limits_{n = 0}^{m}{\gamma }_{L,2n+1},\\{\widetilde{\gamma }}_{2}=\frac{1}{\sqrt{m+1}}\mathop{\sum }\limits_{n = 0}^{m}{\gamma }_{R,2n+1},\end{array}$$
(10)

have exactly zero energy, each delocalized into m + 1 = (N + 1)/2 spatially separated points at the ends of every other stripe. This case is realized in Fig. 2a since \(w{^\prime} \approx 0\) (see Supplementary Fig. 4). The case \(w{\prime} =0\) is realized asymptotically when the stripes become infinitely long. Hence, this case requires one of the system dimensions to be much larger than the other. These two cases are further examples of multi-locational MMs recently predicted to appear in three-terminal Josephson junctions60. In all these cases, the groundstate is twofold degenerate, with two nonlocal MMs \({\widetilde{\gamma }}_{1}\) and \({\widetilde{\gamma }}_{2}\) forming a single zero-energy and particle-hole symmetric fermionic mode. One can thus define two fermionic operators \({Q}_{1,2}={\widetilde{\gamma }}_{1,2}\sqrt{{H}_{{\rm{SUSY}}}}\) satisfying the algebra {P, Qi} = 0, {Qi, Qj} = 2δijHSUSY, where HSUSY is the many-body Hamiltonian with all energy levels positive (obtained by adding a positive constant) and P the fermion parity operator. This corresponds to spontaneously broken \({\mathcal{N}}=2\) quantum mechanical SUSY75 with supercharges Q1,2, zero superpotential, and Witten index W = 1. Furthermore, configurations with several stripes, as in Fig. 2(a), are equivalent to sets of equally-spaced 1D TSs, which can effectively realize the SYK model when coupled to a quantum dot22, or Yang-Lee anyons when coupled to a metallic bath61, provided that the single-particle couplings w and \(w^{\prime}\) are suppressed76. Finally, note that for N → ∞, the two MMs chains at opposite edges form a pair of 1D chiral Majorana edge modes with finite dispersion17,58,62.

Braiding

MMs at the ends of TNSs can be braided. To do so, we need the ability to i) rotate stripes to exchange the MMs and ii) split or merge stripes to implement fusion and read-out. TNSs can be rotated by rotating the in-plane field. Adiabatically decreasing the chemical potential (thus suppressing superconductivity) on specified regions can create domain walls or pointlike defects that split single stripes into two stripes with parallel magnetic fields. The reverse process merges two stripes with parallel fields into one. Figures 2c, d show how to adiabatically split a single stripe into two segments via a domain wall in the middle, and rotate one segment by rotating the in-plane field in one half of the system. Figures 2e, f show how to rotate two stripes around a central pointlike defect. The lowest energy levels corresponding to the MMs remained close to zero in all cases (see Supplementary Figs. 6 and 7). Figure 3a illustrates a possible braiding protocol. Additionally, TNSs can be controlled by moving the domain walls. Alternatively, braiding can be performed in parameter space77,78 without moving the TNSs, e.g., arranging three TNSs around a pointlike defect as in Fig. 3b and controlling the coupling between MMs on opposite ends of the TNSs (see Supplementary Fig. 8).

Fig. 3: Possible braiding protocols.
figure 3

a Rotating left and right stripes exchange the modes γ1, γ2 and γ3, γ4, respectively. Exchanging γ2, γ3 is obtained following a three steps protocol (from top to bottom): i) cut two stripes with antiparallel fields into four (creating four additional Majorana modes), ii) rotate the resulting two inner stripes one around the other, following the arrows, leaving the other two outer stripes unchanged, and iii) join the two inner stripes to the outer stripes. Arrows indicate the direction of the in-plane field. b Braiding can also be performed in parameter space arranging three topologically nontrivial stripes around a pointlike defect and controlling the couplings w1,2,3 between Majorana modes on opposite ends of the stripes, without moving them.

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Discussion

In this work, we described an alternative 2D platform to create, manipulate, and braid MMs via inhomogeneous superconducting orders in proximitized TIs. Unlike other 2D platforms, MMs do not localize at the vortex cores of the order parameter but at the opposite ends of TNSs induced by the inhomogeneous order. This setup can realize topological quantum gates and other exotic quantum phenomena, such as quantum mechanical SUSY, Yang-Lee anyons, and the SYK model. Moreover, TNSs may also be induced by inhomogeneous superconducting orders in Sr2RuO479, iron pnictides80, organic superconductors81,82,83, SrTiO3/LaAlO3 interfaces84, KTaO3/EuO or KTaO3/LaAlO3 interfaces in the inhomogeneous superconducting stripe phase85, and two-component cold atomic Fermi gases with population imbalance and effective spin-orbit coupling62,86,87,88,89,90. Finally, the experimental detection of TNSs would also provide indirect evidence of FFLO inhomogeneous superconductivity.

Methods

The numerical results were obtained by discretizing the continuous Hamiltonian into a lattice model and calculating the energy spectra, wavefunction, and superconducting order parameter self-consistently at zero temperature. The LDOS at zero energy was calculated directly from the energy spectra and wavefunction. The parameters used for the numerical calculations were chosen to be compatible with heterostructures of Bi2Te373 proximitized with NbTiN or NbSe2. The topological invariants (Chern numbers) were calculated numerically using the Fukui-Hatsugai-Suzuki method67, while the parity of the topological invariants was calculated as the sign of the product of the Pfaffians of the Hamiltonian in the Majorana basis with momenta spanning the time-reversal symmetry points in the Brillouin zone.