Quantum theory of magnetic octupole in periodic crystals and application to $d$-wave altermagnets

Magnetic multipoles have been recognized as order parameters characterizing magnetic structure in solids. Recently, magnetic octupoles have been proposed as the order parameters of time-reversal-symmetry breaking centrosymmetric antiferromagnets exhibiting nonrelativistic spin splitting, which is referred to as ``altermagnet’’. However, a gauge-invariant formulation of magnetic octupoles in crystalline solids remains elusive. Here, we present a gauge-invariant expression of spin magnetic octupoles in periodic crystals based on quantum mechanics and thermodynamics, which can be used to quantitatively characterize time-reversal-symmetry breaking antiferromagnets including $d$-wave altermagnets. The allowed physical response tensors are classified beyond symmetry considerations, and direct relationships are established for some of them in insulators at zero temperature. Furthermore, our expression reveals a contribution from an anisotropic magnetic dipole, which has the same symmetry as conventional spin and orbital magnetic dipoles but carries no net magnetization. We discuss the relation between the anisotropic magnetic dipole and the anomalous Hall effect.

💡 Research Summary

The paper addresses a long‑standing problem in the theory of magnetic multipoles in crystalline solids: the lack of a gauge‑invariant formulation for the spin magnetic octupole (MO), which has been proposed as the order parameter of time‑reversal‑symmetry‑breaking, centrosymmetric antiferromagnets (so‑called altermagnets). Starting from the differential form of the free‑energy density, the authors define the spin magnetic dipole (MD), quadrupole (MQ) and octupole (MO) as the linear, first‑order, and second‑order responses to a slowly varying magnetic field and its spatial gradients. By employing thermodynamic identities and the Kubo formula, they derive an auxiliary quantity ˜M_{ijk} and solve the differential equation ∂(βM_{ijk})/∂β = ˜M_{ijk} to obtain a fully gauge‑invariant expression for M_{ijk}.

The final expression (Eqs. 11‑18) shows that M_{ijk} is built from four families of real tensors (g, G, ˜g, ˜G) which are constructed from band‑resolved spin matrix elements s_i^{nm} and velocity matrix elements v_i^{nm}. The diagonal part of M_{ijk} is directly linked to the quantum metric (the real part of the quantum geometric tensor) and to the Berry‑connection polarizability, while the off‑diagonal part involves mixed derivatives in the combined momentum‑magnetic‑field (k‑h) space, reflecting higher‑order geometric quantities such as ∂_h u_n·∂_k u_n. Thus the octupole is not an abstract multipole but a concrete combination of band‑structure geometry.

M_{ijk} is a rank‑3, time‑odd axial tensor with 18 independent components. Using the SO(3) irreducible decomposition, the authors split it into a totally symmetric octupole (M₃), a magnetic toroidal quadrupole (T₂), and two dipolar pieces (M₁ and M₁′). The latter, M₁′, corresponds to the anisotropic magnetic dipole (AMD) originally introduced for isolated atoms. AMD shares the symmetry of ordinary spin and orbital dipoles but carries zero net magnetization; it is defined as M′_i =