Microscopic origin of orbital magnetization in chiral superconductors

TL;DR

This work addresses the microscopic origin of orbital magnetization in time-reversal-breaking chiral superconductors by developing a gauge-invariant framework that couples interband coherence to the intrinsic orbital moment of the Cooper-pair condensate through a dressed photon vertex Γ. The authors derive a general, physically transparent expression for the orbital magnetization Mλ that remains valid beyond the Fermi-surface picture and distinguishes normal-state, mixed normal–Bogoliubov, and Bogoliubov–Bogoliubov contributions. Applying the theory to rhombohedral tetralayer graphene with a phenomenological p-wave pairing, they show that superconductivity can either enhance or suppress the normal-state magnetization depending on Fermi-surface topology, and they uncover a doubly-degenerate generalized clapping mode that gaps due to sublattice winding form factors and renormalizes the electromagnetic vertex. These results lead to concrete experimental signatures, such as magnetization changes measurable by nano-SQUID and quantum oscillations, and establish a framework primed for extensions to moiré superconductors and three-dimensional systems.

Abstract

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 sensitively on the bandstructure. We further identify a generalized clapping mode corresponding to coherent fluctuations between the two opposite chiral windings of the p-wave order parameter, 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. Experimental measurements of the orbital magnetization relative to the quarter-metal phase would provide a direct test of our theory.

Figures (4)

• Figure 1: (a) Schematic illustration of the interband processes contributing to the orbital magnetization in a chiral superconductor, shown for a three-band model consisting of conduction bands $c$ and $c'$, and a valence band $v$. Only the $c$ band is superconducting, while the remaining bands remain in the normal state thus the transitions are classified according to whether the initial and final states are both normal-state quasiparticles (NN), both Bogoliubov quasiparticles (BB), or mixed normal–Bogoliubov states (NB and BN), see Sec. III for further discussion. (b) Orbital magnetization difference, $\Delta M$, between the chiral p-wave superconducting and quarter-metal parent state as the carrier density and interlayer potential is tuned through a Lifshitz transition. The inset shows the Fermi-surface topology of the quarter-metal phase.
• Figure 2: Superconducting ground state of rhombohedral tetralayer graphene in valley $K$ at $V_z=41$ meV and carrier density $n_e=0.5\times10^{12}$ cm$^{-2}$, corresponding to the highest-density point shown in Fig. \ref{['fig:schematic']}b). (a) Fermi-surface contours of $\xi_{c,\mathbf{k}}$ (blue) and $\xi_{c,-\mathbf{k}}$ (orange), together with the self-consistent Bogoliubov--de Gennes spectrum. The black dashed curve shows the normal-state conduction-band dispersion, and the gray dashed horizontal line indicates the Fermi level. (b) Phase of the superconducting order parameter $\Delta_{\mathbf{k}}$. (c) Skyrmion-like texture of the pseudospin field $\mathbf d_{\mathbf k}$, with the in-plane components $(d_x,d_y)$ shown as arrows and the out-of-plane component $d_z$ encoded in the color scale. The in-plane components are normalized by $(d_x^2+d_y^2)^{1/2}$, and $d_z$ is normalized by the total magnitude $d=(d_x^2+d_y^2+d_z^2)^{1/2}$.All calculations are performed using an interaction strength $g=160$ meV$\cdot$nm$^2$, chosen to reproduce the experimentally observed critical temperature.
• Figure 3: Momentum-space distributions of the orbital magnetization contributions from the normal--normal, mixed (NB and BN), and Bogoliubov--Bogoliubov channels: (a) $M_z^{NN}(\mathbf k)$, (b) $M_z^{NN_c}(\mathbf k)$, (c) $M_z^{N_cN}(\mathbf k)$, (d) $M_z^{BB}(\mathbf k)$, (e) $M_z^{NB}(\mathbf k)$, and (f) $M_z^{BN}(\mathbf k)$, shown in units of $\mu_B$ per electron for rhombohedral tetralayer graphene in valley $K$ at $V_z=41$ meV and carrier density $n_e=0.5\times10^{12}$ cm$^{-2}$. The total orbital magnetization obtained by summing over $\mathbf k$ is indicated in the title of each panel. In panel (d), the color scale spans the minimum and maximum values of $M_z^{BB}(\mathbf k)$; note that the peak magnitude is nearly two orders of magnitude larger than the background, so the dominant contribution originates from the region where the electron and hole Fermi surfaces overlap, shown as the dashed white contours. By contrast, the mixed NB and BN contributions arise from the entire Fermi sea rather than being confined to the Fermi-surface region.
• Figure 4: (a) Eigenvalues $\lambda$ of the stability matrix $\mathcal{S}$ (red dots) and $\Delta E_{\bf k}$ (black dash) in Eq. \ref{['eq:S_matrix']}. (b) The Goldstone mode (red arrows) superimposed on top of the groundstate pseudospin distribution (black arrows) (c-d) The pseudospin distribution of the two degenerate generalized clapping modes, which have chirality opposite ($p+ip$) to that of the groundstate ($p-ip$).