Helical Shiba Chain: Topological Superconductivity

• Helical Shiba chains are one-dimensional arrays of magnetic adatoms on s-wave superconductors where impurity moments form a spiral, generating long-range oscillatory Yu–Shiba–Rusinov (YSR) bands.
• The hybridization of YSR states creates an effective p-wave pairing channel that supports topological superconductivity and can harbor Majorana end modes with unconventional localization.
• Defect-induced anti-Shiba states and vacancy spectroscopy provide practical diagnostic tools to assess the robustness and phase transitions in the underlying topological system.

A helical Shiba chain is a one-dimensional array of magnetic adatoms deposited on a conventional $s$-wave superconductor in which the impurity moments form a spiral texture along the chain. Each adatom binds a Yu–Shiba–Rusinov (YSR) state, and the overlap of these bound states produces long-range, oscillatory subgap bands that can realize one-dimensional topological superconductivity and Majorana end modes. In the canonical planar geometry, with adatoms at $x_n = n a$, the texture may be written as

$S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$

so that the helix itself supplies the spin-momentum locking needed for an effective $p$-wave channel, without requiring Rashba spin–orbit coupling in the host (Westström et al., 2016).

1. Geometry, spin texture, and defining mechanisms

The basic object is a chain of classical magnetic moments on a superconducting surface. Several equivalent parametrizations are used in the literature. A planar helix is often written as

$$\mathbf S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$$

while a more general conical texture is

$$\mathbf S_n=\{\sin\theta\cos[\phi(n)],\sin\theta\sin[\phi(n)],\cos\theta\},\qquad \phi(n+1)-\phi(n)=g,$$

with $Q\equiv g$ in lattice units and $\theta=\pi/2$ recovering the in-plane helix (Westström et al., 2016, Samanta et al., 29 Jul 2025). Earlier tight-binding formulations also used a planar $x\text{–}z$ helix,

$$\mathbf B_n = B_0\{\sin(n\theta)\hat x+\cos(n\theta)\hat z\},$$

which is equivalent up to spin-axis conventions (Pöyhönen et al., 2013).

In the helical setting, the magnetic texture mixes spin sectors spatially. The resulting YSR band behaves as a spinless $x_n = n a$0-wave-like superconductor once projected to low energies, which is why Majorana end states can appear. This mechanism is conceptually close to semiconductor-nanowire constructions, but the spin-momentum locking is generated by the helix itself rather than by an externally imposed Rashba term plus Zeeman field (Westström et al., 2016, Pientka et al., 2013). In microscopic discussions of chain formation, the spiral is commonly associated with RKKY exchange, sometimes supplemented by Dzyaloshinskii–Moriya interactions or substrate-induced spin–orbit coupling (Bhowmik et al., 2024).

The helical description is not the only route to the same low-energy structure. A ferromagnetic chain on a Rashba-coupled superconducting surface can generate an analogous topological YSR band because spin-flip correlations in the host Green’s function induce odd-parity pairing even for uniform magnetization (Brydon et al., 2014). This suggests that “helical Shiba chain” denotes both a literal spiral-moment chain and, in a broader effective sense, a class of impurity-band systems whose low-energy YSR sector acquires helical spin structure.

2. Microscopic formulation and effective Shiba-band theory

A standard microscopic starting point is the Bogoliubov–de Gennes Hamiltonian

$x_n = n a$1

with $x_n = n a$2, Nambu matrices $x_n = n a$3, spin matrices $x_n = n a$4, exchange coupling $x_n = n a$5, and $x_n = n a$6-wave gap $x_n = n a$7 (Westström et al., 2016). For a single classical impurity, the YSR energy is

$x_n = n a$8

so the deep-YSR regime corresponds to $x_n = n a$9, where the bound state approaches midgap and hybridization between impurities becomes strong (Westström et al., 2016, Pientka et al., 2013).

In the dilute-chain limit, projection onto the impurity YSR basis yields an effective one-dimensional BdG problem with long-range hopping and pairing,

$S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$0

For a three-dimensional host superconductor, the couplings decay as $S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$1 and oscillate at $S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$2,

$S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$3

with $S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$4 and $S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$5 (Westström et al., 2016). In two-dimensional host models the asymptotics become $S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$6, and the same host-mediated structure produces long-range odd-parity pairing after projection (Brydon et al., 2014, Zhang et al., 2015).

The helical texture enters through spinor overlaps. In the planar case,

$S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$7

so $S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$8-oscillations and helix-induced phases jointly determine the YSR bandwidth and the effective pairing symmetry (Westström et al., 2016). In the deep-dilute regime this reduces to the long-range tight-binding model originally emphasized for helical chains, where hopping and pairing decay only algebraically over distances shorter than the coherence length (Pientka et al., 2013, Pientka et al., 2013).

Beyond the deep-dilute approximation, the subgap problem can be formulated exactly as a nonlinear eigenvalue problem. One form used for helical chains is

$S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$9

with a $p$0 matrix depending nonlinearly on

$p$1

This approach keeps the full energy dependence of the decay length and avoids truncation to short-range couplings (Westström et al., 2016, Westström et al., 2014). Complementary semi-analytical methods treat the infinite chain as a line impurity and extract the effective Green’s function by T-matrix techniques, thereby removing finite-size artifacts from direct tight-binding simulations (Sedlmayr et al., 2021).

3. Topological structure, invariants, and Majorana end states

For a generic helical chain, particle–hole symmetry is present and the system is in class $p$2. For a planar helix, an additional chiral symmetry appears, and the classification is upgraded to class $p$3 (Westström et al., 2016, Pöyhönen et al., 2013). In the planar case the literature frequently uses either a winding number or a $p$4 parity sufficient for the single-Majorana phase. A convenient construction for the exact nonlinear formulation is the “topological Hamiltonian”

$p$5

from which one evaluates a real-space Pfaffian under periodic and antiperiodic boundary conditions; in clean periodic chains, topological transitions coincide with gap closings at $p$6 (Westström et al., 2016). For planar helices, one compact phase-boundary expression is

$p$7

which tracks the change of the $p$8 invariant in the exact treatment (Westström et al., 2016).

In the original long-range Shiba-chain formulation, the Bloch Hamiltonian takes the form

$p$9

with $$\mathbf S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$$0 and $$\mathbf S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$$1 given by slowly decaying oscillatory sums. The planar helix then realizes a long-range analogue of a spinless $$\mathbf S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$$2-wave chain. Unlike the nearest-neighbor Kitaev model, however, both hopping and pairing are power-law over distances $$\mathbf S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$$3, and for nonplanar helices the hopping amplitudes become complex, so $$\mathbf S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$$4 and the pairing between $$\mathbf S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$$5 and $$\mathbf S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$$6 is suppressed (Pientka et al., 2013).

Long-range coupling also alters the critical theory. At the “Bragg point” $$\mathbf S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$$7, the transition between single-channel and two-channel regimes is unconventional: the critical point supports exponentially localized Majorana bound states with a short localization length unrelated to the topological gap, and away from that point the exponential core develops a power-law tail. In the notation of the exact solution,

$$\mathbf S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$$8

while away from criticality the asymptotic tail behaves as $$\mathbf S_n = S[\cos(Qna)\,\hat x + \sin(Qna)\,\hat y],$$9 in the long-coherence-length limit (Pientka et al., 2013). This sharply distinguishes helical Shiba chains from short-range class-$$\mathbf S_n=\{\sin\theta\cos[\phi(n)],\sin\theta\sin[\phi(n)],\cos\theta\},\qquad \phi(n+1)-\phi(n)=g,$$0 wires.

Finite open chains in the topological phase host Majorana zero modes at their ends. In current-biased or FFLO generalizations, these modes remain diagnosed by bulk quantities such as the many-body polarization $$\mathbf S_n=\{\sin\theta\cos[\phi(n)],\sin\theta\sin[\phi(n)],\cos\theta\},\qquad \phi(n+1)-\phi(n)=g,$$1, with $$\mathbf S_n=\{\sin\theta\cos[\phi(n)],\sin\theta\sin[\phi(n)],\cos\theta\},\qquad \phi(n+1)-\phi(n)=g,$$2 marking the topological phase in the self-consistent FFLO studies (Bhowmik et al., 2024). A plausible implication is that “Majorana end mode” remains the unifying observable across static, current-driven, and Floquet helical-chain variants even though the bulk invariants and band constructions differ.

4. Disorder, vacancies, and anti-Shiba physics

The disorder problem in helical Shiba chains is atypical because missing adatoms do not merely broaden the YSR band; they generate their own localized subgap states. In a topological chain, a vacancy acts analogously to a magnetic impurity in a clean $$\mathbf S_n=\{\sin\theta\cos[\phi(n)],\sin\theta\sin[\phi(n)],\cos\theta\},\qquad \phi(n+1)-\phi(n)=g,$$3-wave superconductor and binds a low-lying “anti-Shiba” state below the band edge of the regular chain (Westström et al., 2016). Formally, the defect is treated by a T-matrix,

$$\mathbf S_n=\{\sin\theta\cos[\phi(n)],\sin\theta\sin[\phi(n)],\cos\theta\},\qquad \phi(n+1)-\phi(n)=g,$$4

where $$\mathbf S_n=\{\sin\theta\cos[\phi(n)],\sin\theta\sin[\phi(n)],\cos\theta\},\qquad \phi(n+1)-\phi(n)=g,$$5 is the removal of the local exchange potential and poles of $$\mathbf S_n=\{\sin\theta\cos[\phi(n)],\sin\theta\sin[\phi(n)],\cos\theta\},\qquad \phi(n+1)-\phi(n)=g,$$6 give the vacancy-bound-state energy $$\mathbf S_n=\{\sin\theta\cos[\phi(n)],\sin\theta\sin[\phi(n)],\cos\theta\},\qquad \phi(n+1)-\phi(n)=g,$$7 (Westström et al., 2016).

The physical interpretation is that a missing magnetic atom creates a weak link inside the topological YSR band, binding a localized pair of hybridized Majorana modes. A single vacancy therefore becomes a direct local probe of the nontrivial bulk phase. In the helical chain, vacancy-bound states are generically present throughout the nontrivial $$\mathbf S_n=\{\sin\theta\cos[\phi(n)],\sin\theta\sin[\phi(n)],\cos\theta\},\qquad \phi(n+1)-\phi(n)=g,$$8 region, and their local density of states is concentrated near the defect and decays over $$\mathbf S_n=\{\sin\theta\cos[\phi(n)],\sin\theta\sin[\phi(n)],\cos\theta\},\qquad \phi(n+1)-\phi(n)=g,$$9 with Friedel oscillations at $Q\equiv g$0 (Westström et al., 2016).

At finite vacancy density, the hybridization of these defect states produces an anti-Shiba band. Its bandwidth scales parametrically as

$Q\equiv g$1

up to texture-dependent projector factors, with $Q\equiv g$2 for vacancy concentration $Q\equiv g$3 (Westström et al., 2016). The resulting deterioration of topology is not monotonic in the clean gap. Instead, dilute chains exhibit stripe-like regions of “unusual fragility” in the $Q\equiv g$4 plane whenever a single-vacancy level is tuned close to zero energy. In those regions, even a few percent vacancy concentration can split the nontrivial domain into disconnected islands (Westström et al., 2016).

This vacancy mechanism differs from random fluctuations of the Shiba coupling $Q\equiv g$5. In both ferromagnetic and helical models, $Q\equiv g$6-disorder mainly erodes topology from phase boundaries, whereas vacancy disorder nucleates gapless phases in the middle of otherwise robust topological regions (Westström et al., 2016). This suggests that defect spectroscopy is not only a diagnostic tool but also a stringent constraint on materials design.

5. Broader static variants: ferromagnetic, multichannel, antiferromagnetic, and multichain systems

The helical-chain mechanism belongs to a broader family of YSR-band topological superconductors. One important variant replaces the helical texture by a ferromagnetic chain on a superconducting surface with Rashba spin–orbit coupling. In that setting, odd-parity pairing arises from SOC-induced spin-flip propagators in the host rather than from noncollinear magnetic order, and the effective chain again maps to a spinless $Q\equiv g$7-wave system with long-range couplings (Brydon et al., 2014). Momentum-resolved STM on atom-by-atom Mn chains on Nb(110) later provided evidence for multi-orbital Shiba bands, including a nondegenerate $Q\equiv g$8-derived band with a $Q\equiv g$9-wave gap $\theta=\pi/2$0, while a $\theta=\pi/2$1-derived band remained effectively gapless within the experimental resolution (Schneider et al., 2021).

A second generalization keeps the impurity-band language but includes several angular-momentum scattering channels. In the multichannel YSR chain, simultaneous $\theta=\pi/2$2 and $\theta=\pi/2$3 channels mixed by Rashba SOC generate a multiband Shiba structure. Depending on whether the deep band is $\theta=\pi/2$4-like or $\theta=\pi/2$5-like, the effective theory is either a single-band or two-band topological superconductor, and the inclusion of higher channels can enlarge the topological phase space relative to single-channel models (Zhang et al., 2015). This is directly relevant to realistic transition-metal adatoms, for which multiple YSR resonances are often observed experimentally.

Related static platforms can also be organized around altered magnetic order. In an antiferromagnetic chain on a superconductor, the $\theta=\pi/2$6-pitch spin pattern may be regarded as a special helical limit. There, a supercurrent together with a weak Zeeman field generates a staggered spin-current term and drives the YSR band into a topological phase with spin-filtered Majorana edge states; the edge-spin polarization depends only on the parity of the number of magnetic moments (Heimes et al., 2014). In monolayer transition-metal dichalcogenide superconductors, a ferromagnetic adatom chain can realize the same effective spinless $\theta=\pi/2$7-wave structure through spin–valley locking and parity-mixed intervalley pairing; the topological phase then depends not only on adatom spacing and moment direction but also on the orientation of the chain relative to the crystal axes (Zhang et al., 2016).

The one-dimensional chain can also be coupled laterally into ladders. For planar helices, chirality-reversing domain walls bind two protected Majorana states, and multichain ladders exhibit a transverse-mode structure richer than a simple even–odd $\theta=\pi/2$8 rule. In particular, a ladder of trivial chains can become topological because the transverse couplings renormalize the effective longitudinal parameters of each channel (Pöyhönen et al., 2013). This multichain behavior is one of the clearest indications that helical Shiba systems are better understood as long-range, mode-resolved topological bands than as simple arrays of paired end Majoranas.

6. Supercurrent, FFLO, and Floquet extensions

The static helical Shiba chain has been generalized in two main nonequilibrium directions: phase-biased condensates and periodic drives. A uniform supercurrent in the host superconductor can be encoded by a phase gradient $\theta=\pi/2$9, with $x\text{–}z$0. After a gauge transformation, electrons and holes acquire opposite momentum shifts, and the effective Shiba-chain couplings gain terms linear in $x\text{–}z$1, where $x\text{–}z$2 is the angle between current and chain direction (Röntynen et al., 2014). For nonplanar helices this can tune a chain from gapless to topological gapped, or between trivial and topological gapped phases; for planar helices it mainly proliferates gapless regions (Röntynen et al., 2014).

A more elaborate current-biased extension is the self-consistent FFLO helical Shiba chain, where the order parameter takes the Fulde–Ferrell form

$x\text{–}z$3

With an out-of-plane Zeeman field, the spiral-induced effective SOC and the field-induced band asymmetry stabilize a finite-$x\text{–}z$4 superconducting state that supports end Majorana zero modes and a superconducting diode effect (Bhowmik et al., 2024). In that formulation the bulk topology is diagnosed by the polarization

$x\text{–}z$5

with $x\text{–}z$6 in the topological phase (Bhowmik et al., 2024). A field-free variant proximitizes the helical chain by a $x\text{–}z$7-wave altermagnet, where induced altermagnetic spin splitting stabilizes topological FFLO superconductivity and strong nonreciprocal supercurrents without external fields (Samanta et al., 29 Jul 2025).

Periodic driving introduces a distinct Floquet topology. In the driven helical Shiba chain, a sinusoidal modulation of the chemical potential,

$x\text{–}z$8

creates quasienergy gaps at both $x\text{–}z$9 and $$\mathbf B_n = B_0\{\sin(n\theta)\hat x+\cos(n\theta)\hat z\},$$0, enabling regular $$\mathbf B_n = B_0\{\sin(n\theta)\hat x+\cos(n\theta)\hat z\},$$1-Majorana end modes and anomalous $$\mathbf B_n = B_0\{\sin(n\theta)\hat x+\cos(n\theta)\hat z\},$$2-Majorana end modes (Mondal et al., 2023). Their topology is characterized by dynamical winding numbers computed from periodized evolution operators with twisted boundary conditions (Mondal et al., 2023). Transport theory for the same Floquet Shiba chain shows that the sideband-summed differential conductance obeys a Floquet sum rule, yielding $$\mathbf B_n = B_0\{\sin(n\theta)\hat x+\cos(n\theta)\hat z\},$$3 for $$\mathbf B_n = B_0\{\sin(n\theta)\hat x+\cos(n\theta)\hat z\},$$4 end-localized Floquet Majorana modes at a given quasienergy in the end-resolved limit (Mondal et al., 2024).

These driven and current-biased constructions do not replace the static helical chain; they enlarge its phase space. A plausible synthesis is that the chain has become a flexible platform for engineering effective $$\mathbf B_n = B_0\{\sin(n\theta)\hat x+\cos(n\theta)\hat z\},$$5-wave superconductivity under conditions where the control parameter is no longer only $$\mathbf B_n = B_0\{\sin(n\theta)\hat x+\cos(n\theta)\hat z\},$$6, but also superfluid momentum $$\mathbf B_n = B_0\{\sin(n\theta)\hat x+\cos(n\theta)\hat z\},$$7, drive frequency $$\mathbf B_n = B_0\{\sin(n\theta)\hat x+\cos(n\theta)\hat z\},$$8, and nonequilibrium symmetry breaking.

7. Experimental signatures, materials, and current status

Scanning tunneling microscopy and spectroscopy remain the principal probes. In a conventional static topological phase, one expects end-localized zero-bias peaks separated from the bulk Shiba bands by a minigap, with spatial decay governed by the Majorana localization length. Vacancy spectroscopy adds a complementary bulk-sensitive probe: a single vacancy produces a pronounced LDOS peak at $$\mathbf B_n = B_0\{\sin(n\theta)\hat x+\cos(n\theta)\hat z\},$$9 below the clean band edge, and in fragility stripes this resonance moves close to zero bias (Westström et al., 2016). Two vacancies produce bonding–antibonding splitting that decreases with separation, directly imaging the long-range hybridization of anti-Shiba states (Westström et al., 2016).

The clearest momentum-resolved bulk evidence so far comes from atom-by-atom Mn chains on Nb(110), where Bogoliubov quasiparticle interference revealed multi-orbital Shiba bands and identified one nondegenerate band with a gap shape and particle–hole asymmetry consistent with topological $x_n = n a$00-wave pairing (Schneider et al., 2021). In that system, the fitted $x_n = n a$01-band parameters were

$x_n = n a$02

and the inferred Majorana localization length from the gapped $x_n = n a$03 band was $x_n = n a$04 (Schneider et al., 2021). The same measurements also showed that a gapless $x_n = n a$05 band can complicate the edge phenomenology, underscoring the multiband nature of realistic Shiba chains (Schneider et al., 2021).

Materials proposals span conventional three-dimensional superconductors such as Nb and Pb with adatoms such as Fe, Co, or Mn, atomically assembled chains on heavy-element surfaces with strong Rashba effects, monolayer TMD superconductors, and heterostructures with $x_n = n a$06-wave altermagnets (Schneider et al., 2021, Zhang et al., 2016, Samanta et al., 29 Jul 2025). Across these platforms, the experimentally relevant tuning variables are impurity spacing $x_n = n a$07, effective exchange $x_n = n a$08, helix pitch $x_n = n a$09, chain orientation, and defect density. The collected literature suggests a consistent strategy: maximize the clean topological gap by favorable interference of long-range hopping and pairing, while avoiding vacancy-tuned stripe regions and uncontrolled gapless bands.

Helical Shiba chains therefore occupy a distinctive position within topological-superconductivity research. They are impurity-band systems rather than proximitized semiconductor bands; they are intrinsically long-range rather than nearest-neighbor; and their topology is often most transparent in momentum-resolved bulk observables or defect-bound-state spectroscopy rather than in zero-bias end peaks alone.