The Fundamental Lemma of Altermagnetism: Emergence of Alterferrimagnetism

TL;DR

The paper presents the Fundamental Lemma of Altermagnetism (FLAM), showing that altermagnetic order with zero propagation vector is possible only when a halving subgroup exists in the parent space group and the Wyckoff site-symmetry is contained in that subgroup. It then defines Alterferrimagnetism (AFiM) as a fully compensated ferrimagnetic generalization in which multiple magnetic species can each host AM-like momentum-dependent spin splitting, enabling richer spin textures. Using CuFePO$_5$-type materials as a concrete example, the work demonstrates interpenetrating AM-type sublattices and maps their spin-splitting patterns to specific symmetry constraints. The results provide a practical, symmetry-based framework to identify AM/AFM-like states from crystallographic data and point toward direction-dependent transport and tunable spin polarization for spintronic applications.

Abstract

Recent years have seen a proliferation in investigations on Altermagnetism due to its exciting prospects both from an applications perspective and theoretical standpoint. Traditionally, altermagnets are distinguished from collinear antiferromagnets using the central concept of halving subgroups within the spin space group formalism. In this work, we propose the Fundamental Lemma of Altermagnetism (FLAM) deriving the exact conditions required for the existence of altermagnetic phase in a magnetic material on the basis of site-symmetry groups and halving subgroups for a given crystallographic space group. The spin group formalism further clubs ferrimagnetism with ferromagnetism since the same-spin and opposite-spin sublattices lose their meaning in the presence of multiple magnetic species. As a consequence of FLAM, we further propose a class of fully compensated ferrimagnets, termed as Alterferrimagnets (AFiMs), which can show alternating momentum-dependent spin-polarized non-relativistic electronic bands within the first Brillouin zone. We show that alterferrimagnetism is a generalization of traditional collinear altermagnetism where multiple magnetic species are allowed to coexist forming fully compensated magnetic-sublattices, each with individual up-spin and down-spin sublattices.

Figures (17)

• Figure 1: Hierarchy of conditions required for the existence of altermagnetic (AM) phase in accordance with the fundamental lemma of altermagnetism. For a collinear fully compensated spin arrangement with $\vec{\bm{q}}=0$, if AM phase cannot exist, then antiferromagnetic (AFM) phase will exist. Here, $\mathcal{P}$ is spatial inversion and $t$ is non-trivial centering lattice translation.
• Figure 2: (a) Crystal structure of MFePO$_5$ (M=Fe,Cu,Ni,Co). (b) The coordination polyhedra formed by the different atoms within the crystal structure. (c) The M sublattice shown along with the coordination octahedra with O atoms. The Fe atoms and their coordination octahedra are not shown. (d) The Fe sublattice shown along with the coordination octahedra with O atoms. The M atoms and their coordination octahedra are not shown. In both (c) and (d), the P atoms and their coordination tetrahedra with O atoms have been suppressed for better visualization.
• Figure 3: (a) The Cu sublattice with A-type spin arrangement. The red octahedra denote the up-spin sublattice and the blue octahedra denote the down-spin sublattice. (b) Primitive orthorhombic Brillouin zone (BZ) with the high-symmetry points shown as cyan dots and the irreducible Brillouin zone (IBZ) marked by magenta lines. (c) Spin-polarized electronic total density of states (DOS) of CuGaPO$_5$. (d) Spin-polarized electronic band structure of CuGaPO$_5$ along body diagonals of the IBZ showing spin-splitting. (e) BZ with the $k_x=0$ and $k_x=\pm\frac{\pi}{a}$ planes highlighted in purple. The high-symmetry directions on the respective planes have been highlighted by orange lines and the spin-polarized electronic band structures of CuGaPO$_5$ along these lines have been shown in (f) and (g). (h-j) The same for $k_y=0$ and $k_y=\pm\frac{\pi}{b}$ planes. (k-m) The same for $k_z=0$ and $k_z=\pm\frac{\pi}{c}$ planes. In (e), (h) and (k), the solid orange lines denote paths within the IBZ while the broken orange lines denote paths outside the IBZ.
• Figure 4: (a) The Cu sublattice with C-type spin arrangement. The red octahedra denote the up-spin sublattice and the blue octahedra denote the down-spin sublattice. (b) Primitive orthorhombic Brillouin zone (BZ) with the high-symmetry points shown as cyan dots and the irreducible Brillouin zone (IBZ) marked by magenta lines. (c) Spin-polarized electronic total density of states (DOS) of CuGaPO$_5$. (d) Spin-polarized electronic band structure of CuGaPO$_5$ along body diagonals of the IBZ showing spin-splitting. (e) BZ with the $k_x=0$ and $k_x=\pm\frac{\pi}{a}$ planes highlighted in purple. The high-symmetry directions on the respective planes have been highlighted by orange lines and the spin-polarized electronic band structures of CuGaPO$_5$ along these lines have been shown in (f) and (g). (h-j) The same for $k_y=0$ and $k_y=\pm\frac{\pi}{b}$ planes. (k-m) The same for $k_z=0$ and $k_z=\pm\frac{\pi}{c}$ planes. In (e), (h) and (k), the solid orange lines denote paths within the IBZ while the broken orange lines denote paths outside the IBZ.
• Figure 5: (a) The Cu sublattice with G-type spin arrangement. The red octahedra denote the up-spin sublattice and the blue octahedra denote the down-spin sublattice. (b) Primitive orthorhombic Brillouin zone (BZ) with the high-symmetry points shown as cyan dots and the irreducible Brillouin zone (IBZ) marked by magenta lines. (c) Spin-polarized electronic total density of states (DOS) of CuGaPO$_5$. (d) Spin-polarized electronic band structure of CuGaPO$_5$ along body diagonals of the IBZ showing spin-splitting. (e) BZ with the $k_x=0$ and $k_x=\pm\frac{\pi}{a}$ planes highlighted in purple. The high-symmetry directions on the respective planes have been highlighted by orange lines and the spin-polarized electronic band structures of CuGaPO$_5$ along these lines have been shown in (f) and (g). (h-j) The same for $k_y=0$ and $k_y=\pm\frac{\pi}{b}$ planes. (k-m) The same for $k_z=0$ and $k_z=\pm\frac{\pi}{c}$ planes. In (e), (h) and (k), the solid orange lines denote paths within the IBZ while the broken orange lines denote paths outside the IBZ.

Theorems & Definitions (2)

• Lemma
• proof