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

TL;DR

The authors derive a gauge-invariant quantum-mechanical expression for the spin magnetic octupole $M_{ijk}$ in periodic crystals and connect it to thermodynamic response tensors via Maxwell relations, enabling a systematic description of time-reversal-symmetry breaking antiferromagnets including $d$-wave altermagnets. They decompose $M_{ijk}$ into MO, MTQ, and two magnetic-dipole components, revealing the anisotropic magnetic dipole (AMD) as a distinct, non-magnetization-carrying contribution and linking AMD to the anomalous Hall effect in altermagnetic models. Through model calculations on collinear AFM/FM systems (MnF$_2$-like and RuO$_2$-like), they demonstrate how MO components respond to exchange, SOC, and band-structure details, and show thermodynamic relations that tie MO to quadrupolar magnetoelectric couplings and higher-rank AHC terms. The work provides a unified framework to characterize TRSB magnets beyond symmetry alone, with AMD offering a concrete link to experimentally accessible probes such as XMCD and the AHE. Temperature and SOC effects further modulate MO, suggesting avenues for exploring nonrelativistic altermagnetic physics in real materials.

Abstract

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.

Figures (17)

• Figure 1: Electronic band dispersions of the collinear magnetic models. Band structures in the paramagnetic (a), the antiferromagnetic (b), and the ferromagnetic states (c) without SOC calculated from the Hamiltonian in Eqs. \ref{['eq:Hk_for_Para']}, \ref{['eq:Hk_for_AFM']}, and \ref{['eq:Hk_for_FM']}, respectively. The parameters are chosen to reproduce the paramagnetic band structures of MnF$_2$ obtained in Ref. roig:prb2024 (see also Methods section for details) and $J=0.1$ in (b) and (c).
• Figure 2: Finite components of $M_{ijk}$ as a function of $J$. [(a) and (b)] ([(c) and (d)]) $J$ dependence of the nonzero components of $M_{ijk}$ for $\bm{J}=(J,0,0)$ ($\bm{J}=(0,0,J)$) in the AFM and FM, respectively. We choose $\lambda=0.1$ and $T=0.01$.
• Figure 3: Finite components of $M_{ijk}$ as a function of $\lambda$. [(a) and (b)] ([(c) and (d)]) $\lambda$ dependence of the nonzero components of $M_{ijk}$ when $\bm{J}=(J,0,0)$ ($\bm{J}=(0,0,J)$) in the AFM and FM, respectively. We choose $J=0.1$ and $T=0.01$.
• Figure 4: Chemical potential dependence of the MO. (a) Antiferromagnetic band structure of the collinear model with an insulating gap. Unlike Fig. \ref{['fig:band']}, the offset due to the Fermi level is not taken into account. (b) $M_{xxy}$ is plotted as a function of the chemical potential. (c) Enlarged view of the boxed area in (b). In (a)-(c), the parameters are chosen to be as follows: $\bm{J}=(J,0,0)$, $\bm{\lambda}_{\bm{k}}=(\lambda_{x,\bm{k}},0,0)$, $J=0.5$, and $\lambda=0.1$. We choose $T=0.01$ in (b) and (c). The shaded areas correspond to the energy gap.
• Figure 5: Magnetic dipoles as a function of $J$.$J$ dependence of the finite components of the AMD ($M'_{i}$) and the spin magnetization ($M_{i}$) in the AFM (FM) is shown in (a) ((b)), where the vertical axis is expressed in units of $\mu_{\mathrm{B}}$. Although the spin magnetization remains zero in the AFM, it is shown for reference. The parameters are the same as those in Fig. \ref{['fig:Mijk_jdep']}.