Spin Symmetry Criteria for Odd-parity Magnets

TL;DR

Using a spin space group framework, the work derives symmetry criteria for the emergence of odd-parity magnets (OPMs) in systems with negligible spin-orbit coupling. OPMs are organized into eight symmetry-driven cases and categorized by Bloch-state spin textures into type-I (collinear), type-II (coplanar), and type-III (noncoplanar), with additional symmetry constraints on $p$-wave and $f$-wave spin-splitting for type-I. It identifies 48 candidate materials in the Magndata database that satisfy these symmetry criteria and validates the criteria with two minimal lattice constructions H1 (coplanar triangular) and H2 (noncoplanar square), illustrating type-I and type-II OPMs. It demonstrates that OPMs can host intrinsic $\mathbb{Z}_2$ topology, exemplified by a bilayer breathing Kagome lattice via an effective time-reversal symmetry $\tilde{T}$ with $\tilde{T}^2 = -1$, and confirms this with Wilson-loop calculations and edge-state spectra.

Abstract

Inspired by the discovery of altermagnets, which exhibit even-parity nonrelativistic spin splitting, odd-parity magnets (OPMs) have been proposed and emerged as a novel research frontier. In this study, we perform a comprehensive spin group symmetry analysis to establish symmetry criteria for the emergence of OPMs. We identify eight distinct symmetry-driven cases that support OPMs, enabling their realization in collinear, coplanar, and noncoplanar magnetic orders. These OPMs are categorized into three types based on their spin textures for Bloch states: collinear (type-I), coplanar (type-II), and noncoplanar (type-III). For type-I OPMs, we further delineate additional symmetry requirements for $p$-wave and $f$-wave spin splitting. We identify 48 candidate materials in the Magndata database that satisfy these symmetry criteria. Additionally, we construct two theoretical models to validate the effectiveness of the established symmetry criteria. Finally, we show that OPMs can exhibit an intrinsic $\mathbb{Z}_2$ topology and construct a theoretical model to realize this phase.

Figures (2)

• Figure 1: (a) Schematic illustration of the $120^{\circ}$ antiferromagnetic state on a triangle lattice with the $\sqrt{3}\times \sqrt{3}$ structures. (b) The isoenergy-surface characterized by $s_z(\bm k)$ for the model shown in (a). (c) Unit cell for a 2D lattice model on a square lattice. The vectors $(x,y,z)$ denotes the magnetic moment directions. (d) The isoenergy-surface characterized by the vector $(s_x(\bm k,s_y(\bm k))$, shown by the blue arrow, for the lattice model in (c). We take $t=J=1$, and $\mu=-2.5$ for (b) and $\mu=-3.2$ for (d).
• Figure 2: (a) Schematic illustration of a bilayer kagome breathing lattice. (b) The wannier center $W(k_y)$ for the lowest energies two bands as a function of $k_y$. (c) The energy spectrum for the system with a nanowire geometry along the $x$ direction. The red and blue bands denote the edge states from opposite edges. (d) The spin polarization Fermi contour. The color encodes the value $s_z(\bm k)$. The pink arrows denote the direction of the vector $(s_x(\bm k), s_y(\bm k))$ at $\bm k$. In (b-d), $t_{\perp}=t_1=1$ and $t_2=J=4$. In (d) we take $\mu=-3.3$