Topological superconductivity in skyrmion lattices

The study addresses how to create, control, and manipulate 2D topological superconducting phases in magnet–superconductor hybrid systems to realize and study Majorana zero modes and chiral Majorana edge modes. While 2D MSH systems are predicted to possess rich Chern-number phase diagrams, experimental tunability between distinct phases has been lacking. The authors propose and analyze 2D MSH systems with a magnetic skyrmion lattice, showing that varying the skyrmion radius—tunable via external magnetic field—enables transitions between different topological superconducting phases. The mechanism is linked to a spatially inhomogeneous Rashba spin–orbit interaction induced by the skyrmion texture, which carries a measurable signature in the local density of states and can be probed by Josephson scanning tunneling spectroscopy (JSTS).

The paper builds on observations of Majorana modes in 1D and 2D topological superconductors and the use of MSH systems (chains, islands, layers of magnetic adatoms on s-wave superconductors) as platforms for topological superconductivity and STS studies. Prior theoretical work predicted high-Chern-number phases in 2D ferromagnetic Shiba lattices and detailed their topological phase diagrams. Creation of Majorana modes in single skyrmions has been discussed previously, and classification places such systems in class D (broken time-reversal, particle–hole symmetry). Real-space topological markers (Chern number density) have been developed to map topology in inhomogeneous systems. JSTS techniques have been advanced to image superconducting order parameters and supercurrents at the atomic scale. Recent related efforts showed that engineered non-collinear magnetic textures can produce 1D topological superconductivity and Majorana modes; this work extends such ideas to 2D skyrmion lattices.

The authors study a 2D magnet–superconductor hybrid consisting of a magnetic skyrmion lattice deposited on an s-wave superconductor. Electrons reside on a triangular lattice (lattice constant a0) with nearest-neighbor hopping t, chemical potential μ, and s-wave pairing Δ. Local moments Sr (classical, magnitude S) couple via exchange J to the conduction electrons. The Hamiltonian is formulated in real and Nambu space. The skyrmion lattice is taken commensurate with the substrate, with radius R in integer or half-integer multiples of a0 and a triangular arrangement. A unitary transformation rotates local spins Sr to the z-axis, mapping the texture to an out-of-plane ferromagnetic order plus a spatially inhomogeneous, effective Rashba spin–orbit (RSO) coupling α(r). The induced α(r) mirrors the local skyrmion number density ns(r) (local topological charge), peaking at skyrmion centers and vanishing at unit-cell corners, with an overall scale ∝ 1/R. Topological characterization uses the Chern number C computed via the projector formalism in momentum space (sum over occupied bands) and, in parallel, the real-space Chern number density C(r), which averages to the same C. The electronic structure is obtained from the retarded Green’s function G(ω) = [(ω + iδ)I − H]−1 to calculate the local (spin-resolved) density of states N(r, ω, σ). Supercurrents are computed within the Keldysh formalism. Triplet pairing correlations and the critical Josephson current are evaluated following established JSTS theory, including simulations with s-wave and triplet-paired tips. Representative parameter sets include (μ, Δ, JS) = (−5, 0.4, 0.5)t for maps at R ≈ 500 Å, and a ribbon case with (JS, Δ, μ) = (0.5, 0.4, 5.5)t and R = 5a0 producing C = 3.

- Skyrmion-induced inhomogeneous RSO coupling enables tunable 2D topological superconductivity: changing the skyrmion radius R (via external magnetic field) drives transitions between distinct Chern-number phases. - Phase boundaries follow μ ≈ A + B/R^2, indicating that α(r) ∝ 1/R effectively renormalizes the chemical potential and controls topological transitions alongside μ and J. - The phase diagrams in (μ, R), (μ, JS) for fixed R, and (JS, R) for fixed μ reveal that for sufficiently large JS, incremental changes of R by half a lattice constant can change the system’s Chern number, enabling access to a rich set of phases. - Real-space topological marker: C(r) peaks at unit-cell corners where α(r) is minimal, while ns(r) and α(r) peak at skyrmion centers. Lowest-energy states concentrate where α is small, providing a real-space analog of Berry curvature localization in momentum space. - At a representative topological phase transition (e.g., C = 8 to C = 6), the bulk gap closes at K/K′ points with Dirac cones; the zero-energy LDOS is maximal at unit-cell corners (where α is minimal) and minimal at skyrmion centers (where α is maximal). The change in Chern number equals the multiplicity m of gap-closing momenta: m = 1 (Γ), 3 (M), 2 (K/K′), or 6 (non-TRI points along M lines), yielding ΔC = m when such closings occur. - Skyrmion ribbons in topological phases exhibit C chiral Majorana edge modes per edge, linearly dispersing across the gap; zero-energy LDOS localizes along edges with spatial modulation complementary to ns(r). In regions of large α (skyrmion centers), LDOS weight is pushed to higher energies. - Broken time-reversal symmetry induces supercurrents that circulate not only along ribbon edges but also around each skyrmion inside the ribbon, akin to screening currents in vortex lattices. - The skyrmion texture induces spin-triplet pairing correlations in equal-spin (↑↑, ↓↓) and mixed-spin (Sz = 0) channels. JSTS with appropriately engineered s-wave or triplet tips can image the spatial structure of the s-wave order parameter suppression Δ(r) and the real/imaginary parts of nonlocal triplet correlations, respectively, providing an experimental handle on a key ingredient of topological superconductivity.

The work demonstrates a concrete and experimentally accessible route to tune 2D topological superconducting phases in MSH systems by controlling skyrmion radius with magnetic fields well below typical upper critical fields. The induced, spatially varying RSO coupling from the skyrmion texture is the key mechanism enabling this tunability, effectively shifting phase boundaries and allowing controlled changes in Chern number. Real- and momentum-space analyses are consistent: LDOS patterns at transitions and along edges mirror the distribution of α(r) and the local topological charge. The presence of chiral Majorana edge modes satisfies bulk–boundary correspondence and provides signatures in spectroscopic maps. Critically, JSTS can visualize induced spin-triplet correlations that are necessary for topological superconductivity, suggesting an experimental diagnostic for identifying topological phases in such systems. Overall, the findings advance quantum engineering strategies for manipulating Majorana modes and exploring topology-driven transport and edge phenomena in 2D settings.

The paper introduces skyrmion-lattice MSH systems as tunable 2D topological superconductors where varying the skyrmion radius controls transitions among high-Chern-number phases. It identifies the induced, spatially inhomogeneous Rashba spin–orbit interaction as the mechanism for tunability, establishes real-space signatures in LDOS and Chern number density, and demonstrates chiral Majorana edge modes and distinctive supercurrent patterns. It further proposes JSTS as a direct probe of induced spin-triplet correlations, offering a pathway to experimentally identify topological superconductivity. Future research directions include: experimental realization and JSTS imaging of triplet correlations in skyrmion MSH systems; systematic mapping of phase diagrams versus R, μ, and JS; exploration of other non-collinear 2D magnetic textures as tunable platforms; and investigations of disorder, interaction effects beyond classical spin approximations, and skyrmion profile changes under varying fields.

The study is theoretical and assumes classical, static local moments with a fully gapped s-wave superconductor (neglecting quantum spin fluctuations and Kondo screening). It presumes a commensurate triangular skyrmion lattice matching the substrate and no intrinsic Rashba coupling. The precise phase diagram depends on the skyrmion radial profile and its field dependence. Finite-size and idealized boundary conditions are used for ribbons. Experimental verification, material-specific effects, and robustness to disorder or incommensurability are not fully addressed.