Microscopic origin of orbital magnetization in chiral superconductors

Chiral superconductivity is a time-reversal-symmetry-breaking superconducting phase that has attracted broad interest as a potential platform for topological quantum computation. A fundamental consequence of this symmetry breaking is orbital magnetization, yet a clear microscopic formulation of this quantity has remained elusive. This difficulty arises because Bogoliubov quasiparticles do not carry a definite electric charge, precluding a simple interpretation of orbital magnetization in terms of circulating quasiparticle currents. Moreover, superconductivity and ferromagnetism rarely coexist, and in the few materials where they do (e.g. uranium-based compounds), strong spin-orbit coupling obscures the orbital contribution to the magnetization. The recent report of chiral superconductivity in rhombohedral multilayer graphene, which has negligible spin-orbit coupling, therefore provides a unique opportunity to develop and test a microscopic theory of orbital magnetization in chiral superconductors. Here we develop such a theory, unifying the interband coherence effects underlying normal-state orbital magnetization with the intrinsic orbital moments of the Cooper-pair condensate. Applying our theory to rhombohedral tetralayer graphene, we find that the onset of superconductivity can either enhance or suppress the normal-state orbital magnetization, depending on the bandstructure. We further identify a generalized clapping mode with a gap set by the sublattice winding form factor. This collective mode is unique to chiral superconductors and contributes to the orbital magnetization through its role in dressing the photon vertex. Our theory resolves a long-standing conceptual difficulty in defining orbital magnetization in superconducting systems, and measurements of the orbital magnetization relative to the quarter-metal phase would provide a direct experimental test.

💡 Research Summary

The paper tackles a long‑standing problem in the theory of chiral (time‑reversal‑symmetry‑breaking) superconductors: how to define and calculate orbital magnetization when the elementary excitations are Bogoliubov quasiparticles that carry no definite electric charge. The authors begin with a generic k·p continuum Hamiltonian for electrons in a crystal, introduce minimal coupling to a vector potential, and formulate the problem in Nambu space. By constructing the BCS mean‑field Hamiltonian and the associated photon vertex, they show that in the normal (non‑superconducting) state the quasiparticle velocity operator coincides with the bare photon vertex, reflecting charge conservation for Landau quasiparticles. In the superconducting state, however, the dressed photon vertex Γ_k differs from the quasiparticle group‑velocity operator because Bogoliubov quasiparticles are charge‑neutral superpositions of electrons and holes.

Through a linear‑response treatment of the grand potential Ω in the presence of a slowly varying magnetic field, they derive a compact expression for the orbital magnetization (Eq. 12, later refined to Eq. 14). The final formula involves a double sum over occupied (i) and unoccupied (m) quasiparticle states, weighted by (E_m+E_i)/(E_m−E_i)^2 and the antisymmetric tensor ε_λμν. Two distinct matrix elements appear: the dressed photon vertex ⟨U_m|Γ_ν|U_i⟩, which encodes interband coherence and many‑body renormalization, and the derivative of the BCS Hamiltonian ⟨U_i|∂_μ H_BCS|U_m⟩, which captures the intrinsic orbital moment of Cooper pairs. This structure unifies the normal‑state orbital magnetization (originating from Berry curvature and interband coherence) with the superconducting contribution (intrinsic pair orbital moment), while respecting gauge invariance and charge‑conservation constraints.

The authors classify the total magnetization into four parts—NN (normal‑state contribution), NB and BN (normal‑Bogoliubov mixing), and BB (pure Cooper‑pair contribution)—and emphasize that superconductivity often involves only a subset of bands, making the NB/BN terms particularly important. They then apply the formalism to rhombohedral tetralayer graphene, a system with negligible spin‑orbit coupling and a well‑characterized quarter‑metal phase that already exhibits a large anomalous Hall effect and measurable orbital magnetization via nano‑SQUID.

Using a phenomenological attractive p‑wave interaction projected onto the lowest conduction band, they solve the self‑consistent BCS gap equations and obtain the Bogoliubov‑de Gennes spectrum. Two distinct scenarios emerge depending on the Fermi‑surface topology of the quarter‑metal parent: (i) when three disjoint Fermi pockets exist, the onset of superconductivity enhances the orbital magnetization; (ii) when the Fermi surface is a single connected contour, superconductivity suppresses the magnetization. This sensitivity to band topology provides a clear experimental signature: combined nano‑SQUID and quantum‑oscillation measurements across the Lifshitz transition should reveal a sign change in ΔM = M_S – M_QM.

A further novel result is the identification of a generalized “clapping mode”—coherent fluctuations that flip the chirality of the p‑wave order parameter. The mode’s gap is set by a sublattice winding form factor, and its coupling to the electromagnetic field dresses the photon vertex, thereby contributing an additional term to the orbital magnetization. Because this mode is unique to chiral superconductors, its detection (e.g., via THz spectroscopy or Raman scattering) would constitute direct evidence of intrinsic chiral pairing.

In summary, the paper delivers a rigorous microscopic theory of orbital magnetization in chiral superconductors that integrates interband Berry‑phase effects with Cooper‑pair orbital moments, respects fundamental symmetries, and yields concrete, experimentally testable predictions for graphene‑based chiral superconductors. The work resolves a conceptual impasse that has persisted for decades and opens a pathway to quantitatively probe the orbital component of magnetism in topological superconducting states.