Chiral superconductivity stabilized by hole doping

Introduction

Recent theoretical work shows that introducing holes into a Chern ferromagnet can give rise to an unconventional superconducting state. By using the repulsive Kane‑Mele‑Hubbard model, researchers demonstrated that two doped holes can bind to a single spin‑flip excitation (a magnon) and form a composite Cooper pair. This pair carries charge 2e, carries a finite spin chirality, and exhibits chiral f‑wave symmetry. Crucially, the interaction between these skyrmion‑bipolaron pairs switches from repulsive to attractive when the Ising spin‑orbit coupling (SOC) exceeds a narrow threshold, creating a window where the pairs can condense into a chiral superconducting phase.

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1. Skyrmion‑Bipolaron Pairing Mechanism

• Composite Cooper pair – The bound state consists of two holes and one magnon (2h1s). It is a charge‑2e boson with spin projection \(S_z=\pm2\) and a non‑coplanar skyrmionic spin texture inherited from the parent Chern ferromagnet.
• Chiral f‑wave symmetry – The spatial part of the pair wavefunction transforms as an f‑wave representation, directly linked to the Chern number of the undoped system.
• Spin chirality as a fingerprint – Finite spin chirality signals the non‑trivial topology of the underlying ferromagnet and distinguishes this pairing from conventional electron‑electron Cooper pairs.

The binding arises from spin‑flip excitations that lower the kinetic energy of the doped holes. Unlike earlier magnonic superconductivity proposals, no external magnetic field is required; the skyrmion‑bipolaron forms spontaneously at zero field because the Chern ferromagnet already carries a net polarization.

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2. Role of Ising Spin‑Orbit Coupling

Density‑matrix‑renormalization‑group (DMRG) calculations reveal a critical SOC range:

• For \(\lambda < \lambda_{c,1}\) (≈ 0.222 for \(U=100\)), the effective interaction between skyrmion‑bipolarons remains repulsive.
• Between \(\lambda{c,1}\) and \(\lambda{c,2}\) (≈ 0.23), the interaction becomes attractive, allowing the dilute bosons to condense.
• Above \(\lambda_{c,2}\) the attraction weakens again, shrinking the viable window.

Although the intermediate‑SOC regime is narrow, its existence within the paradigmatic Kane‑Mele‑Hubbard model suggests a degree of robustness that could survive modest variations in material parameters.

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3. Connection to MoTe₂ Moiré Superlattices

The theoretical phase diagram aligns with experimental reports of superconductivity in hole‑doped MoTe₂ moiré superlattices. In those systems, strong Ising SOC and a Chern‑type band structure are present, matching the conditions identified for skyrmion‑bipolaron attraction. The f‑wave symmetry and spin‑chirality predicted here provide concrete signatures that can be probed via tunneling spectroscopy or phase‑sensitive Josephson interferometry.

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4. Numerical Approach and Validation

• Exact diagonalization on a 12‑unit‑cell cluster captured the essential two‑hole‑one‑magnon physics.
• DMRG on quasi‑1D ribbons (up to 24 sites) allowed systematic scaling of the interaction strength \(U\) and SOC \(\lambda\).
• Boundary pinning fields were employed to prevent artificial localization of magnons at the edges, ensuring bulk‑like behavior.
• Convergence was monitored via entanglement entropy, particle number, and energy, with sweeps halted once relative errors fell below a preset threshold.

These complementary methods give confidence that the observed repulsive‑to‑attractive transition is not a numerical artifact but a genuine many‑body effect of the model.

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Practical Outlook

The identification of a concrete microscopic mechanism—hole‑magnon binding with chiral f‑wave symmetry—opens several experimental pathways:

1. Material engineering – Search for Chern ferromagnets with tunable Ising SOC (e.g., transition‑metal dichalcogenide moiré heterostructures) where hole doping can be controlled electrostatically.
2. Spectroscopic probes – Look for signatures of finite spin chirality in neutron scattering or resonant X‑ray techniques.
3. Phase‑sensitive measurements – Use corner‑junction interferometry to test for f‑wave order parameter symmetry.
4. Cold‑atom analogues – Simulate the Kane‑Mele‑Hubbard model with synthetic SOC and controlled hole doping to directly observe skyrmion‑bipolaron formation.

By targeting the narrow SOC window identified theoretically, experimentalists can assess whether the predicted skyrmion‑bipolaron condensation indeed underlies the observed chiral superconductivity in moiré systems.

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Conclusion

Hole doping a Chern ferromagnet creates a novel Cooper pair composed of two holes and a magnon, whose chiral f‑wave symmetry and spin chirality are dictated by the parent topological band. When Ising spin‑orbit coupling lies within a modest intermediate range, the interaction between these pairs becomes attractive, enabling a condensate that manifests as chiral superconductivity. This mechanism provides a concrete, experimentally testable route to unconventional superconductivity and may explain recent observations in MoTe₂ moiré superlattices. Future work should focus on material platforms that allow precise SOC tuning and on probes that can directly detect the predicted spin‑chirality and f‑wave order.