Noncollinear Magnetic Multipoles in Collinear Altermagnets

TL;DR

This work addresses real-space magnetic multipoles in altermagnets, showing that even collinear magnets host noncollinear spin-density components due to spin-orbit coupling. By combining group theory with density functional theory, it reveals that higher-order multipoles, including 32-poles, can be nonzero and sometimes dominate certain observables in MnF$_2$, CrSb, and KMnF$_3$, and that structural distortions can induce or control altermagnetic order. The study connects multipole content to measurable tensor observables, such as piezomagnetism and second-order magnetoelectric effects, highlighting the role of SOC and lattice symmetry in shaping macroscopic responses. Overall, NSD and higher-order multipoles, modulated by SOC and structural degrees of freedom, provide a robust framework for understanding and predicting altermagnetic behavior in real materials.

Abstract

Altermagnets host an array of magnetic multipoles, which are often visualized and studied in the reciprocal space. In the real space, the relative phase of the multipoles of the spin-density around atoms determines whether a system is an altermagnet or a conventional antiferromagnet. In this study, we approach these real space multipoles in altermagnets using a combination of first principles calculations and group theory. We show that even in collinear magnets, the local spin density is necessarily noncollinear due to spin-orbit coupling. Moreover, the noncollinear contributions often provide a more direct illustration of the magnetic multipolar character of altermagnetism than the collinear contribution, which is dominated by the dipolar term. Our first principles calculations also show that 32-poles, in addition to the octupoles, can be visible in spin-density of d-wave altermagnets, and they must be taken into account in discussions of the macroscopic response. Finally, we elucidate the interplay between magnetism and subtle crystal structural distortions in perovskite altermagnets, which provide a fertile playground for studying phase transitions between antiferromagnetic and altermagnetic phases.

Figures (5)

• Figure 1: (a) Crystal and magnetic structures of MnF$_2$ and CrSb. The blue lines represent the screw axes, the red and green arrows represents the local $x$, $y$ coordinate axes, and the black arrows represent the spins. (b) Isosurfaces of collinear spin density $m_z(\mathbf{r})$ around a magnetic ion obtained from DFT in the absence of SOC. When the SOC is introduced, the atomic dipoles do not tilt, but a nodal spin density is induced in the $m_x(\mathbf{r})$, $m_y(\mathbf{r})$ components, displayed in (c), with a smaller isosurface value than (b).
• Figure 2: (a) The isosurfaces for different components of DFT spin densities of MnF$_2$. Orange and blue colors correspond to positive and negative values, and the isosurface values used for collinear ($m_z$) and noncollinear ($m_x$ and $m_y$) are not equal. (b) Examples of decompositon onto the tesseral harmonics of the spin density in MnF$_2$. The atomic magnetic octupoles have the units of $\mu_Ba_0^2$, while the 32-poles have the units of $\mu_Ba_0^4$, where $\mu_B$ is the Bohr magneton and $a_0$ is the Bohr radius. Red and blue data correspond to the Mn atoms on the corners and the body-center of the unit cell, which are equal for IP multipoles (e.g. $d_{xy}m_z$) and opposite for OP multipoles (e.g. $m_z$ and $d_{xz}m_x+d_{yz}m_y$). Results for the rest of the multipoles are reported in the SM Supplement.
• Figure 3: (a) The isosurfaces for different components of DFT spin densities of CrSb. Orange and blue colors correspond to positive and negative values, and the isosurface values used for collinear ($m_z$) and noncollinear ($m_x$ and $m_y$) are not equal. (b) Examples of decompositon onto the tesseral harmonics of the spin density in CrSb. The atomic magnetic octupoles have the units of $\mu_Ba_0^2$, while the 32-poles have the units of $\mu_Ba_0^4$, where $\mu_B$ is the Bohr magneton and $a_0$ is the Bohr radius. Red and blue data correspond to the Cr atoms on the edge and the body-center of the unit cell, which are equal for IP multipoles (e.g. $d_{xy}m_x+d_{x^2-y^2}m_y$) and opposite for OP multipoles (e.g. $d_{z^2}m_z$). Results for the rest of the multipoles are reported in the SM Supplement.
• Figure 4: (a) The $m_z(\mathbf{r})$ spin density around the Mn ions in KMnF$_3$ is aligned with the F octahedra, and rotates without much distortion along with the octahedra. F ions are shown as grey spheres. (b) While the integrated $sm_z$ atomic dipole density remains almost constant as a function of the octahedral rotation angle, the IP 32-poles $g_{xy(x^2-y^2)}m_z$ turn on linearly with octahedral rotations. All other multipole components are reported in the SM Supplement.
• Figure 5: The dependence of the piezomagnetic tensor components to the corresponding magnetic octupoles in MnF$_2$. Different data points are generated by repeating the DFT calculations for different SOC strengths. (a) $\Lambda_{xyz}$, which relates the $z$ component of magnetization to $xy$ shear strain, is symmetry allowed even in the absence of SOC, however, it is forced to be zero by Luttinger compensation at zero temperature since there is a gap. However, the corresponding atomic octupole $d_{xy}M_z$ is not affected by the compensation and is nonzero for all values of SOC. The interplay between the compensation and response leads to a nontrivial dependence of this component of the piezomagnetic tensor on the atomic multipole strength. (b) For small values of SOC, the $\Lambda_{xzy}$ component, which relates the $y$ component of magnetization to $xz$ shear strain, linearly depends on the $d_{xz}M_y$ magnetic octupole, and both quantities are zero in the absence of SOC.