High-performance descriptor for magnetic materials: Accurate discrimination of magnetic structure

The magnetic structure is crucial in determining the physical properties inherent in magnetic compounds. We present an adequate descriptor for magnetic structure with proper magnetic symmetry and high discrimination performance, which does not depend on artificial choices for coordinate origin, axis, and magnetic unit cell in crystal. We extend the formalism called “smooth overlap of atomic positions” (SOAP), providing a numerical representation of atomic configurations to that of magnetic moment configurations. We introduce the descriptor in terms of the vector spherical harmonics to describe a magnetic moment configuration and partial spectra from the expansion coefficients. We discuss that the lowest-order partial spectrum is insufficient to discriminate the magnetic structures with different magnetic anisotropy, and a higher-order partial spectrum is required in general to differentiate detailed magnetic structures on the same atomic configuration. We then introduce the fourth-order partial spectrum and evaluate the discrimination performance for different magnetic structures, mainly focusing on the difference in magnetic symmetry. The modified partial spectra that are defined not to reflect the difference of magnetic anisotropy are also useful in evaluating magnetic structures obtained from the first-principles calculations performed without spin-orbit coupling. We apply the present method to the symmetry-classified magnetic structures for the crystals of ${\mathrm{Mn}}_3\mathrm{Ir}$ and ${\mathrm{Mn}}_3\mathrm{Sn}$, which are known to exhibit anomalous transport under the antiferromagnetic order, and examine the discrimination performance of the descriptor for different magnetic structures on the same crystal.

Figure 1

The $r_{\mathrm{cut}}$ dependence of the similarity kernels for the ferromagnetic (FM) and antiferromagnetic (AFM) systems. (a)–(d) show the $K^{\left(2\right)}$ and $K^{\left(4\right)}$ values as a function of ${\ell }_{\max }$ with fixed $n_{\max }=6$ or $n_{\max }$ with fixed ${\ell }_{\max }=7$. The magnetic moments are ordered along the $z$ axis on each atomic site in the one-dimensional periodic lattice with 1 Å interval along the $x$ axis.

Figure 2

Similarity kernels with different values of $\sigma$, using $n_{\max }=6$ and ${\ell }_{\max }=7$ for FM and AFM configurations on the one-dimensional crystal, (a) $K^{\left(2\right)}$ and (b) $K^{\left(4\right)}$. (c) and (d) compare the similarity kernels for FM and different $\mathrm{AFM}i$ magnetic patterns, where the magnetic moments have opposite signs per $i$-atomic sites. $\sigma =1.0\AA , n_{\max }=6$, and ${\ell }_{\max }=7$ are used.

Figure 3

The spin rotation dependence of the normalized similarity kernels between the magnetization density of the ferromagnetic structure along the (100) axis, denoted by $\mathit{m}$, and those of the rotated magnetic moments, given by $R_s(\theta ,\varphi )\mathit{m}$. The results are presented for (a) $K^{\left(2\right)}$ and (b) $K^{\left(4\right)}$ on a simple BCC lattice with lattice constant $a=1.0$ Å, (c) $K^{\left(4\right)}$ on a body center tetragonal lattice with $a=1.0$ Å and $c=1.5$ Å, and (d) $K^{\left(4\right)}$ on a hexagonal lattice with $a=1.0$ Å and $c=1.5$ Å, using $n_{\max }=4, {\ell }_{\max }=6$, and $r_{\mathrm{cut}}=5.0$ Å.

Figure 4

Symmetrized magnetic structures generated on ${\mathrm{Mn}}_3\mathrm{Ir}$ crystal by cluster multipole method with magnetic moment normalized to unity [17].

Figure 5

Symmetrized magnetic structures generated on ${\mathrm{Mn}}_3\mathrm{Sn}$ crystal by cluster multipole method with each finite magnetic moment normalized to unity [17].

Figure 6

Pair plots of normalized similarity kernels of the power spectrum and trispectrum with $\sigma =4.0$ Å, $n_{\max }=4, {\ell }_{\max }=6$, and $r_{\mathrm{cut}}=5$ Å for the magnetic structures corresponding to Fig. 4 for (a) ${\overline{K}}^{\left(2\right)}$, (b) $K^{\left(2\right)}$, (c) ${\overline{K}}^{\left(4\right)}$, and (d) $K^{\left(4\right)}$.

Figure 7

Pair plots of normalized similarity kernels of the power spectrum and trispectrum with $\sigma =4.0$ Å, $n_{\max }=4, {\ell }_{\max }=6$, and $r_{\mathrm{cut}}=5$ Å for magnetic structures corresponding to Fig. 5 for (a) ${\overline{K}}^{\left(2\right)}$, (b) $K^{\left(2\right)}$, (c) ${\overline{K}}^{\left(4\right)}$, and (d) $K^{\left(4\right)}$.

Figure 8

Pair plots of normalized similarity kernel $K^{\left(4\right)}$ between the symmetrized magnetic structures of ${\mathrm{Mn}}_3\mathrm{Ir}$ and of ${\mathrm{Mn}}_3\mathrm{Sn}$, shown in Figs. 4 and 5. The calculations use $n_{\max }=4, {\ell }_{\max }=6$, and $r_{\mathrm{cut}}=5$ Å with (a) $\sigma =1.0$ and (b) 4.0 Å.