Orbital Magnetism in Chiral Superconductors

• Orbital magnetization in chiral superconductors is a phenomenon where spontaneous time-reversal symmetry breaking and nonzero Cooper pair angular momentum generate net magnetic moments.
• Modern Berry-curvature and BdG approaches reveal the complex interplay between condensate intrinsic moments and interband coherence, ensuring gauge-invariant results.
• Finite-size scaling, screening effects, and spin-orbit coupling critically shape edge currents and experimental signatures in both mesoscopic and bulk regimes.

Orbital magnetization in chiral superconductors refers to the net orbital magnetic moment generated by the spontaneous breaking of time-reversal symmetry and the presence of topologically protected chiral edge modes. Unlike conventional superconductors, chiral superconductors feature pairing states that carry nonzero angular momentum, giving rise to unique bulk and edge magnetization phenomena driven by the condensate’s internal degrees of freedom and the structure of the superconducting gap. The precise understanding and measurement of orbital magnetization in such systems is central to identifying signatures of chiral pairing, distinguishing its microscopic origin, and interpreting magneto-optical and SQUID-detected phenomena.

1. Fundamental Formulation of Orbital Magnetization in Chiral Superconductors

In the microscopic framework, the orbital magnetization ($M_z$) in superconductors arises as the thermodynamic derivative of the grand potential $\Omega$ with respect to an externally applied magnetic field: $$M_\lambda = -\frac{1}{A} \left.\frac{\partial \Omega}{\partial B_\lambda}\right|_{B\to 0}$$ where $A$ is the sample area and $B=\nabla\times \mathbf{A}$ the magnetic field (Zhu et al., 18 Jan 2026). In mean-field BCS theory, $\Omega$’s response to small fields can be evaluated using the linear response of the BdG quasiparticle density matrix, manifesting as mixing between BdG eigenstates through the current (photon-vertex) operator.

The modern Berry-curvature theory of orbital magnetization, originally formulated for normal metals, has been generalized to superconductors within the Bogoliubov–de Gennes (BdG) formalism (Robbins et al., 2018, Zhu et al., 18 Jan 2026). The physical complexity arises because BdG quasiparticles lack a definite electric charge and are not eigenstates of current; thus, orbital magnetization depends on the intricate interplay between the BdG photon vertex and the velocity operator.

The resulting gauge-invariant expression for orbital magnetization in 2D is: $$M_\lambda = \frac{e}{4} \int_{\text{BZ}} \frac{d^2k}{(2\pi)^2} \sum_{m \in \text{unocc},\,i \in \text{occ}} \frac{E_{mk} + E_{ik}}{(E_{mk} - E_{ik})^2} \epsilon_{\lambda\mu\nu} \,\mathrm{Im}\left[ \langle U_{mk} | \Gamma_k^\nu | U_{ik}\rangle\, \langle U_{ik} | \partial_\mu H^{BCS}(k) | U_{mk} \rangle \right]$$ where $|U_{nk}\rangle$ are BdG eigenstates and $E_{nk}$ their energies, $\Gamma_k^\nu$ the photon vertex, and $\partial_\mu H^{BCS}(k)$ the band velocity (Zhu et al., 18 Jan 2026). This formula recovers the normal-state Berry-curvature result when $\Delta \rightarrow 0$.

2. Microscopic Mechanisms: Cooper Pair Angular Momentum and Momentum-Space Magnetism

Orbital magnetization in chiral superconductors receives contributions from two physically distinct microscopic mechanisms:

• Normal-State Interband Coherence: Superconductivity modifies the coherent interband transitions that give rise to orbital magnetization in the normal state. In the limit of vanishing gap, the usual Berry-curvature expressions for normal metals are recovered (Zhu et al., 18 Jan 2026).
• Condensate Intrinsic Orbital Moment: Chiral pairing with nonzero angular momentum (e.g., $d+id$, $p+ip$) produces a direct “condensate” magnetization, which is absent in $s$-wave states. The dominant term originates from the $d(k) \times d^*(k)$ structure of the pairing, encoding the Cooper pair’s internal angular momentum (Geng et al., 22 Jun 2025), and survives even in the absence of spin-orbit coupling.

Spin-orbit coupling and mixed-parity states introduce additional momentum-space “magnetism,” yielding both out-of-plane (polar) and in-plane orbital magnetization textures (Geng et al., 22 Jun 2025). The total local orbital magnetization density is: $$M_{\mathrm{orb}}(\mathbf{k}) = i\delta_0 [E+A_0(\mathbf{k})][d(\mathbf{k})\times d^*(\mathbf{k})]_z + i\delta_0 [A(\mathbf{k})\times (d_0(\mathbf{k})d^*(\mathbf{k})-d_0^*(\mathbf{k})d(\mathbf{k}))]_z$$ where $\delta_0$ is a resonance denominator, $A(\mathbf{k})$ is the spin-orbit coupling vector, and $d_0$, $d$ refer to singlet and triplet amplitudes (Geng et al., 22 Jun 2025).

3. Quasiclassical and Mesoscopic Effects: Size Scaling and Edge Currents

In finite-sized mesoscopic chiral $d$-wave superconductors, the orbital magnetization and chiral edge current exhibit nontrivial scaling with system size (Holmvall et al., 2023). The self-consistent quasiclassical formalism (Eilenberger equations) determines spatially inhomogeneous edge currents and the associated orbital moment: $$\mathbf{J}(\mathbf{r}) = 2eT\sum_{n}^{|\varepsilon_n|<\omega_c}\langle \mathbf{v}_F(\mathbf{p}_F)g(\mathbf{r},\mathbf{p}_F;\varepsilon_n)\rangle_{p_F}$$

$$m_z = \frac{m_0}{2}\int_{\mathcal{A}}\frac{d^2\mathbf{r}}{\mathcal{A}} \left[\mathbf{r}\times\frac{\mathbf{J}(\mathbf{r})}{j_0}\right]_z$$

with $j_0 = \hbar |e|^2/\xi_0$ and $m_0 = N\hbar|e|/m^*$ (Holmvall et al., 2023).

For large disks of radius $\mathcal{R} \gg \xi_0$, both the total chiral edge current $I(\mathcal{R})$ and orbital magnetic moment $m(\mathcal{R})$ scale as $1/\mathcal{R}$. This scaling fails for small $\mathcal{R} \lesssim 20\,\xi_0$, where edge–edge hybridization dramatically enhances $I$ and $m$, leading to pronounced local maxima and sign reversal of the net current for $\mathcal{R}\lesssim 7\,\xi_0$ (Holmvall et al., 2023).

4. Role of Screening, Topology, and Collective Modes

Meissner screening modifies the spatial structure and net value of edge currents and orbital magnetization. For $\lambda_0 \gg \mathcal{R}$, screening is negligible; for $\lambda_0 \lesssim \mathcal{R}$, the Meissner response re-arranges the edge current profile, suppressing the net $I$ and $m$, though the induced local flux and $B$-field remain finite. The trapped flux thus serves as an alternative robust experimental signature even when net current vanishes (Holmvall et al., 2023).

The interplay between condensate angular momentum, collective modes, and topological edge phenomena is essential. In chiral $p$-wave superconductors, a generalized “clapping mode” corresponding to coherent fluctuations between $p+ip$ and $p-ip$ windings modifies the effective photon vertex (vertex corrections), which must be included for quantitative accuracy in $M_z$ calculations (Zhu et al., 18 Jan 2026). The splitting of the clapping mode is determined by sublattice-winding form factors and is observable as a resonance in electromagnetic response, particularly in rhombohedral tetralayer graphene systems.

5. Lattice Models, Berry Curvature, and Gauge Considerations

In periodic systems, direct use of the position operator $\hat{\mathbf{r}}$ is ill-defined, motivating Berry-curvature-based (modern theory) approaches (Robbins et al., 2018). Within the BdG framework, the orbital magnetization divides into “local circulation” and “itinerant circulation” terms—extensions of their normal-state counterparts—built from cell-periodic Bloch components of the BdG wavefunctions: $$\mathbf{M}_{\mathrm{LC}} = -\gamma\,\mathrm{Im}\sum_n \int_{\mathrm{BZ}}\frac{d^3k}{(2\pi)^3} \left[ \langle \partial_k u_{nk}| \times \hat{H}_k | \partial_k u_{nk} \rangle f(E_{nk}) - \langle \partial_k v_{nk}| \times \hat{H}_k^* | \partial_k v_{nk} \rangle (1-f(E_{nk})) \right]$$

$$\mathbf{M}_{\mathrm{IC}} = \gamma\,\mathrm{Im}\sum_n \int_{\mathrm{BZ}}\frac{d^3k}{(2\pi)^3} E_{nk} \left[ \langle \partial_k u_{nk}| \times | \partial_k u_{nk} \rangle f(E_{nk}) + \langle \partial_k v_{nk}| \times |\partial_k v_{nk} \rangle (1 - f(E_{nk})) \right]$$

with $$\mathbf{M} = \mathbf{M}_{\mathrm{LC}} + \mathbf{M}_{\mathrm{IC}}$$ (Robbins et al., 2018).

In realistic multiband systems (e.g., Sr₂RuO₄), strong cancellations between these contributions and the on-site orbital moment render $M_z$ orders of magnitude smaller than naive single-band predictions, explaining why edge currents and orbital magnetization are below current experimental sensitivity (Robbins et al., 2018).

Gauge invariance is essential: Each component, $\mathbf{M}_{\mathrm{LC}}$ or $\mathbf{M}_{\mathrm{IC}}$, is not individually gauge invariant, but their sum is. The structure of the gauge transformation in Nambu space precludes the simple separation of “occupied” and “unoccupied” states familiar from normal-state theory.

6. Experimental Signatures and Measurement Strategies

Key experimental implications differ substantially between the bulk and mesoscopic regimes. In macroscopic chiral superconductors, orbital magnetization and associated stray fields are predicted to be extremely small, well below leading-edge magnetometry resolution ($\lesssim 1\,\mu$G for Sr₂RuO₄) (Robbins et al., 2018). In contrast, mesoscopic disks ($R\sim 10$–$20\,\xi_0$) of chiral $d$-wave or $p$-wave materials show dramatic enhancement of both edge current and orbital moment—even at low $T$—with stray fields of $10^{-4}$–$10^{-3}$ T readily detectable by nano-SQUID or scanning probes (Holmvall et al., 2023).

A crucial prediction in rhombohedral multilayer graphene is that the sign of the change in orbital magnetization at the superconducting transition depends sensitively on carrier density and interlayer bias (Zhu et al., 18 Jan 2026). Observation of a generalized clapping collective mode at $\Delta_\mathrm{clap}\sim 0.1$–$0.3$ meV in in-plane polarized Raman or microwave spectroscopy would serve as direct evidence for intrinsic chiral pairing and its momentum-space structure (Zhu et al., 18 Jan 2026).

The optical anomalous Hall effect in chiral superconductors is tightly connected to the momentum-space orbital magnetization: the Kerr and analogous responses depend on symmetry-allowed combinations of the pairing-induced magnetization density in the Brillouin zone, as determined by generalized Onsager relations (Geng et al., 22 Jun 2025). The physical origin can be either out-of-plane Cooper pair angular momentum (from non-unitary gaps) or in-plane texture (from spin-orbit- and mixed-parity-induced “unitary” contributions).

7. Summary Table: Key Mechanisms and Regimes

Regime / Mechanism	Dominant Contribution	Detectable Signal
Macroscopic, periodic bulk	Cancellation of bulk + edge moments	$M_z\lesssim 10^{-6}\,\mu_B$; stray field $\ll 1\,\mu$G (Robbins et al., 2018)
Mesoscopic finite size ($R\sim10$–$20\,\xi_0$)	Edge–edge hybridization, finite size enhancement, condensate moment	Edge currents $I\sim 0.1\,I_0$, $B$-field $10^{-4}$–$10^{-3}\,$T (Holmvall et al., 2023)
Low/High Carrier Density Graphene	Change in $M_z$ by band topology	Sign change in $\Delta M$, clapping collective mode (Zhu et al., 18 Jan 2026)
Optical (Hall/Kerr) Response	Momentum-space orbital magnetization from $d\times d^*$ and SOC	Polar/in-plane Kerr rotation, frequency-dependent $\sigma_{xy}$ (Geng et al., 22 Jun 2025)

• Enhanced chiral edge currents and orbital magnetic moment in chiral $d$-wave superconductors from mesoscopic finite-size effects (Holmvall et al., 2023)
• Modern Theory for the Orbital Moment in a Superconductor (Robbins et al., 2018)
• Microscopic origin of orbital magnetization in chiral superconductors (Zhu et al., 18 Jan 2026)
• Pairing-induced Momentum-space Magnetism and Its Implication In Optical Anomalous Hall Effect In Chiral Superconductors (Geng et al., 22 Jun 2025)