Collinear Altermagnets and their Landau Theories

TL;DR

The paper develops a symmetry-based Landau theory for altermagnets, first in the ideal zero-SOC limit and then with finite SOC, to connect spin-split band patterns with symmetry-allowed observable responses. It catalogs all $\mathbf{Q}=\mathbf{0}$ altermagnets through a 54-case classification based on 1D real irreps of the crystal point group, introduces secondary multipolar order parameters that couple to the Néel vector, and establishes a concrete criterion linking SOC-free multipoles to SOC-coupled observables. The finite-SOC analysis yields a practical coupling rule: a tensor observable $\xi$ couples linearly to the Néel vector if $\Gamma_{\xi}$ lies in $\Gamma_{\xi} \otimes (aeV \otimes \Gamma_{\mathbf{N}})$, enabling predictions of transport and optical responses (e.g., AHC, magnetoresistance, THC, Kerr, piezomagnetism) across point groups. The methodology is illustrated with multiple material examples (CrF$_2$, La$_2$CuO$_4$, MnF$_2$, MnTe, Fe$_2$O$_3$, MnTe), providing symmetry-based guidance for identifying and characterizing altermagnetic phases in real compounds.

Abstract

Altermagnets exhibit spontaneously spin-split electronic bands in the zero spin-orbit coupling (SOC) limit arising from the presence of collinear compensated magnetic order. The distinctive magneto-crystalline symmetries of altermagnets ensure that these spin splittings have a characteristic anisotropy in crystal momentum space. These systems have attracted a great deal of interest due to their potential for applications in spintronics. In this paper, we provide a general Landau theory that encompasses all three-dimensional altermagnets where the magnetic order does not enlarge the unit cell. We identify all crystal structures that admit altermagnetism and then reduce these to a relatively small set of distinct possible Landau theories governing such systems. In the zero SOC limit, we determine the possible local multipolar orders that are tied to the spin splitting of the band structure. We make precise the connection between altermagnetism as defined at zero SOC ("ideal" altermagnets) and the effects of weak SOC. In particular, we examine which response functions allowed by symmetry when SOC is present are guaranteed by the spin-orbit free theory, and spell out the distinctive properties of altermagnets in comparison with conventional collinear antiferromagnets. Finally, we show how these ideas can be applied by considering a number of altermagnetic candidate materials.

Figures (6)

• Figure 1: Illustrations of all possible altermagnetic spin-splitting anisotropies in momentum space allowed by symmetry. These correspond to the spatial anisotropies of the lowest order multipole that can couple to the aletermagnetic order parameter ${\bf N}$.
• Figure 2: The crystal structure of MnTe with space group symmetry $P6_{3}/mmc$. Magnetic Mn ions (red and blue denote magnetic sublattices) reside on the $2a$ Wyckoff positions, at $\{0,0,0\}$ and $\{0, 0, \frac{1}{2}\}$ within the unit cell. The Te ions (gray) occupy the Wyckoff positios $4e$, at $\{\frac{1}{3},\frac{2}{3},\frac{1}{4}\}$, and $\{\frac{2}{3},\frac{1}{3},\frac{3}{4}\}$.
• Figure 3: The crystal structure of CrF$_{2}$ with space group symmetry $P2_{1}/c$. We use the setting $P1\,2_{1}/n\,1,$ related to the original setting by $\{\mathbf{a},\mathbf{b},\mathbf{c}\}\rightarrow\{-\mathbf{a}-\mathbf{c},\mathbf{b},\mathbf{a}\}.$ Magnetic Cr ions (red and blue denote magnetic sublattices) reside on the $2b$ Wyckoff positions, at $\{0,0,0\}$ and $\{\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\}$ within the unit cell. The F ions (gray) occupy the Wyckoff positions $4e$, at $\pm \{x,y,z\}$, and $\pm \{x+\frac{1}{2},\frac{1}{2}-y,z+\frac{1}{2}\}$, forming a distorted octahedral environment tilting out of the $bc$ plane.
• Figure 4: La$_{2}$CuO$_{4}$ structure and magnetic sublattices. The space group is $\mathbf{G} = Bmab$ (No. 64). This setting is related to $Cmce$ by $\mathbf{c} \leftrightarrow -\mathbf{b},$ and has a pure half-translation $\{\frac{1}{2},0,\frac{1}{2}\}$. Magnetic Cu atoms (red and blue) occupy the $4a$ WP $\{0,0,0\}$ and $\{\frac{1}{2},\frac{1}{2}, 0\}.$ La atoms (cyan) reside on the $8f$ WP $\pm \{x,y,0\}$, $\pm \{x + \frac{1}{2}, -y+\frac{1}{2},0\}$. O atoms (grey) occupy two WP, $8f$ and $8e$, at $\{x,\frac{1}{4},\frac{1}{4}\}$, $\{x+\frac{1}{2}, \frac{1}{4},\frac{1}{4}\}$, $\{-x,\frac{3}{4},\frac{3}{4}\}$, $\{-x+\frac{1}{2}, \frac{3}{4}, \frac{3}{4}\}$).
• Figure 5: The crystal and magnetic sublattice structure of MnF$_{2}$, with space group $P4_{2}/mnm$ (No. 136). Mn atoms (red and blue denote magnetic sublattices) reside on the $2a$ WP $\{0,0,0\}$ and $\{\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\}$, while F atoms (grey) occupy the $4f$ WP with positions $\pm\{x,x,0\}$ and $\pm\{-x+\frac{1}{2},x+\frac{1}{2},\frac{1}{2}\}$.