Thermodynamic formulation of the spin magnetic octupole moment in bulk crystals

TL;DR

This paper resolves the fundamental challenge of defining the spin magnetic octupole moment (SMOM) in bulk crystals by formulating a thermodynamic, gauge-invariant description based on the gradient expansion of the grand potential. The authors derive a Brillouin-zone expression for the SMOM that remains well-defined without the unbounded position operator, establish Středa-type relations linking SMOM to spin magnetoelectric dipole-quadrupole susceptibilities, and verify these relations numerically in several lattice models. In particular, they demonstrate that nonrelativistic SMOM components can dominate in $d$-wave altermagnets, sharing a microscopic origin with nonrelativistic spin splitting and displaying Néel-vector dependent behavior consistent with Landau theory. The results provide a practical, first-principles-friendly framework for computing SMOM in real materials and suggest experimental avenues, such as neutron scattering and magnetoelectric-response measurements, to access SMOM and related octupolar order parameters. Overall, the work links high-rank magnetic multipoles to tangible bulk responses and offers new routes to probe exotic altermagnetic phases.

Abstract

The discovery of unconventional antiferromagnets, such as altermagnets, has drawn significant attention to higher-rank magnetic multipoles, particularly magnetic octupoles. Despite the advances in research, attempts to understand their microscopic properties remain limited due to the unbounded nature of the position operator in bulk crystals. In this paper, we address this problem by using a well-known thermodynamic approach and derive a formula for the spin magnetic octupole moment (SMOM) that can be used in bulk crystals. The resulting formula is gauge invariant and satisfies Středa formulas that relate the SMOM to the spin magnetoelectric dipole-quadrupole susceptibilities. Furthermore, we apply this formula to several models and examine the fundamental properties of the SMOM. For example, in $d$-wave altermagnets, the nonrelativistic component of the SMOM, which is independent of spin-orbit coupling, is larger than the relativistic component, which is induced by spin-orbit coupling. These nonrelativistic SMOMs have the same microscopic origin as the nonrelativistic spin splitting that characterizes $d$-wave altermagnetism. Moreover, they exhibit a Néel vector dependence consistent with Landau theory for $d$-wave altermagnetism [Phys. Rev. Lett. $\textbf{132}$, 176702 (2024)].

Figures (6)

• Figure 1: Band dispersions of the 2D altermagnet model along high-symmetry lines for (a) $N/N_c=0.5$, (b) $N/N_c=1.0$, and (c) $N/N_c=1.25$. The red and blue lines represent the up-spin and down-spin bands, respectively. The region of $N/N_c<1$ is a nontrivial regime, and $N/N_c>1$ is a trivial regime, where the $X$ and $Y$ points are gap-closing points. Furthermore, this model exhibits spin degeneracy along the $k_x=\pm k_y$ lines (e.g., the $\Gamma$-$M$ line), which reflects $d_{x^2-y^2}$ symmetry. Here, we use the same parameters as in the numerical calculations and set $W=8t_d$.
• Figure 2: (a) Band dispersion of the pyrochlore model for $(t,m)=(1.0,0.5)$ without SOC (dashed line) and with SOC (solid line). This model has Weyl points around $E=2.5$ in the presence of SOC, as indicated by the red arrow. (b) Chemical potential dependence of the SMOM without the SOC (red line) and with the SOC (blue line). A noticeable enhancement does not appear around $\mu=2.5$. (c) Magnified view of (a) with SOC. An insulating phase exists in the region of $0.75 \le E \le 0.85$ (red-shaded area). (d) Chemical potential dependence of the SMOM in the insulating region at several temperatures $T$. In the numerical calculations, the other parameters are set to $\lambda=0.1$ in (d) and $(t,m)=(1.0,0.5)$ in both (b) and (d).
• Figure 3: (a) Chemical potential dependence of the SMOM in the 2D altermagnet model for $N/N_c=0.5$ (red line) and $N/N_c=1.25$ (blue line). (b) $N$ dependence of the slope ($\partial O_{z,xx}/\partial \mu$). The inset is a magnified view of the region highlighted in the main panel. The sign of the slope changes within this range.
• Figure 4: (a) Chemical potential dependence of the SMOM at $N/N_c=0.99$ calculated by decomposing Eq. \ref{['eq:octupole']} into each term. The labels "$^{(2)}$G. P. D." and "F. S." indicate the contributions from a two-band term proportional to the grand potential density and the Fermi sea terms, respectively. The other terms are denoted by "Others", and the total contribution is represented by "Total" note1. (b) Momentum-resolved quantity of $\partial O_{z,xx}/\partial\mu$ at $N/N_c=0.99$.
• Figure 5: $J$ dependence (a) and SOC dependence (b) of the SMOM in the 3D altermagnet model. We set $\lambda=0.1$ in (a), $J=0.2$ in (b), and $\mu=0.25$ in both panels.