First-principles theory of spin magnetic multipole moments in antiferromagnets

Abstract

Antiferromagnets with vanishing net magnetization are naturally expected to host higher-order magnetic multipole moments. Understanding and utilizing the multipole degrees of freedom are imperative for novel conceptual designs and applications unique to antiferromagnets. However, a universal, quantitative definition of magnetic multipole moments of antiferromagnetic materials is currently lacking. In this work we provide a unified description of arbitrary-order spin magnetic multipole moments (SM$^3$) of antiferromagnets by introducing a nonlocal spin density in macroscopic Maxwell equations. The formalism makes it transparent how SM$^3$ calculated for translationally invariant bulk systems corresponds to experimental observables when translation symmetry is broken. Through the nonlocal spin density calculated from first principles, we propose a robust scheme to extract arbitrary-order SM$^3$ through symmetry-constrained fitting at long wavelengths. Using this approach, we have calculated SM$^3$ of a few representative antiferromagnets, including $α$-$\rm Fe_2O_3$, Mn$_3$Sn, and Mn$_3$NiN. Moreover, we clarify the role of spin-orbit coupling (SOC) in SM$^3$, especially in the weak SOC limit where clean predictions can be made based on symmetry principles. Our work paves the way for systematically investigating multipolar order parameters of unconventional magnetic materials.

Figures (7)

• Figure 1: Schematic illustration of spin densities (blue arrows) created by a spatially nonuniform magnetic octupole for the toy model in Sec. \ref{['sec:model']}. The curved surface corresponds to a random $\psi(\mathbf r)$ in Eq. \ref{['eq:psirM']}. The scale of the kagome lattice is exaggerated for illustration purpose.
• Figure 2: (a) Crystal structure and magnetic order of the toy model. (b) Band structure of the model. $t = 1, t_{\rm SO} = 0.5, J = -5$. The horizontal solid and dashed lines represent $\mu = 0$ (insulator) and $\mu = -3.5$ (metal), respectively.
• Figure 3: (a) Integrand of $\chi^{x}(\mathbf q)$ plotted along the high-symmetry path as in Fig. \ref{['fig:model']} (b) for the insulating case and an arbitrarily chosen $\mathbf q = (0.02,0.08,-0.03)$ (in units of $1/a$). (b) $\chi^x (\mathbf q)$ calculated on a grid near the Brillouin zone center (orange dots) and its approximant (blue surface) using the fitted octupole moments plotted on the $k_y = 0$ plane.
• Figure 4: (a) Integrand of $\chi^{x}(\mathbf q)$ plotted along the high-symmetry path as in Fig. \ref{['fig:model']} (b) for the metallic case and a smallest nonzero $\mathbf q$ on a $31\times 31\times 31$ mesh. (b) Same as (a) but for a smallest nonzero $\mathbf q$ on a $150\times 150\times 150$ mesh and $k_B T = 0.01$. (c) $\chi^x (\mathbf q)$ calculated on the $31\times 31\times 31$ mesh near the Brillouin zone center (orange dots) and its approximant (blue surface) using the fitted octupole moments plotted on the $k_y = 0$ plane.
• Figure 5: (a) Structure and magnetic order of $\alpha$-$\rm Fe_2O_3$ in the canted antiferromagnet state. (b) $\chi^x (\mathbf q)$ calculated on a $20\times 20\times 20$ mesh near the Brillouin zone center (orange dots) and its approximant (blue surface) using the fitted octupole moments plotted in the $(010)$ plane.