Cartesian Nodal Lines and Magnetic Kramers Weyl Nodes in Spin-Split Antiferromagnets

TL;DR

This work shows that spin-split antiferromagnets can host two distinct, symmetry-protected band-degeneracy types: Cartesian nodal lines that arise without SOC and magnetic Kramers Weyl nodes that persist with SOC. Through a concrete tight-binding model, it demonstrates how mirror and $C_{4z}\mathcal{T}$-type symmetries pin these degeneracies and regulate their Berry-curvature distributions, leading to topological boundary states with unconventional spin textures. The authors predict strong or quantized anomalous Hall effects under weak Zeeman fields and quantized circular photogalvanic effects when Weyl nodes are suitably arranged, highlighting potential experimental platforms. Materials such as MnTe$_2$, Mn$_3$IrGe, and YMnO$_3$ are proposed as candidates, with spin-resolved ARPES and CPGE/MOKE measurements offering routes to observe the novel CNL and MKWN physics and their associated boundary phenomena.

Abstract

When band degeneracy occurs in a spin-split band structure, it gives rise to divergent Berry curvature and distinctive topological boundary states, resulting in a variety of fascinating effects. We show that three-dimensional spin-split antiferromagnets, characterized by symmetry-constrained momentum-dependent spin splitting and zero net magnetization, can host two unique forms of symmetry-protected band degeneracy: Cartesian nodal lines in the absence of spin-orbit coupling, and magnetic Kramers Weyl nodes when spin-orbit coupling is present. Remarkably, these band degeneracies not only produce unique patterns of Berry-curvature distributions but also give rise to topological boundary states with unconventional spin textures. Furthermore, we find that these band degeneracies can lead to strong or even quantized anomalous Hall effects and quantized circular photogalvanic effects under appropriate conditions. Our study suggests that spin-split antiferromagnets provide a fertile ground for exploring unconventional topological phases.

Figures (8)

• Figure 1: Schematic of the lattice structure. White (blue) spheres represent nonmagnetic (magnetic) atoms, while red arrows indicate the local magnetic moments. [(a)] presents the magnetic background surrounding a representative nonmagnetic atom; the configurations around other nonmagnetic atoms are identical and thus omitted for clarity. (b) Schematic illustration of the Wigner-Seitz unit cell. The spin (red) and real-space (black) coordinate conventions are the same as in [(a)]. (c-e) Schematics of the spin-dependent hoppings of the form $\boldsymbol{S}\cdot\boldsymbol{\sigma}$ in the $xy$, $xz$, and $yz$ planes, respectively. Dashed black arrows indicate hopping directions, and the corresponding amplitudes are shown in light-blue rounded rectangles. Panels [(d,e)] share the same spin-coordinate convention as [(c)].
• Figure 2: Cartesian nodal lines and their associated topological properties. (a) Solid and dashed orange lines represent nodal lines, which form a structure analogous to the Cartesian coordinate system at each intersection. (b) Distribution of the Berry curvature in the momentum plane with $k_{z}=0.1$. (c) Energy spectra along a path in the surface Brillouin zone, with open boundary conditions in the $[\bar{1}10]$ direction. High-symmetry points in the surface Brillouin zone are shown in the inset. (d) Black arrows and gradient colors jointly depict the spin polarizations of the surface states on the right $(\bar{1}10)$ surface. The yellow network marks the boundary of the zero-energy surface states. Common parameters are $t=t_{z}=\lambda_{\rm so}=0$, and $\lambda_{\rm M}=\eta=1$.
• Figure 3: Magnetic Kramers Weyl nodes and their associated topological surface states. (a) Schematic of MKWNs in Brillouin zone when $\eta>\eta_{c}$. Blue and red spheres represent MKWNs with opposite charges, illustrated in the middle inset. (b) Fermi arcs on the left (right) $x$-normal surface is denoted by solid (dotted) purple lines, and the black arrows denote their spin polarizations. The solid green rings represent the projections of bulk states at fixed energy $E=0.2$. Parameters are $t=t_{z}=0$, $\lambda_{\rm M}=0.4$, $\lambda_{\rm so}=1.1$, and $\eta=1$.
• Figure 4: Anomalous Hall effects activated by a $z$-direction Zeeman field. (a) Two representative band-degeneracy configurations giving rise to Hall plateaus under a Zeeman field applied along the $z$ direction. Left: $\lambda_{\rm so}=0$, where the $z$-direction Zeeman field deforms the CNLs into two nodal rings. Right: $0<\lambda_{\rm so}<B_{z}$, where four pairs of Weyl nodes (spheres in red and blue) remain whose $k_{z}$ components are fixed and independent of $B_{z}$. In both panels, the light red (light blue) regions mark the range of $k_z$ for which each $k_z$ slice is a Chern insulator with $\mathcal{C}=1$ ($\mathcal{C}=-1$). (b) Evolution of the Hall conductivity tensor $\sigma_{xy}$ (in units of $e^{2}/h$) as a function of $B_{z}$. Parameters are $t=t_{z}=0$, $\mu=0$, and $\lambda_{\rm M}=\eta=1$.
• Figure 5: Quantized circular photogalvanic effect. (a) Band structures along high-symmetry lines of the Brillouin zone for different values of anisotropy parameter $\eta$ in the magnetic exchange field. The two choices of Fermi energy are $\mu_{1}=-2t-t_{z}$ and $\mu_{2}=-2t+t_{z}$. (b) Trace of the CPGE tensor $\mathrm{Tr}[\beta(\omega)]$ (in units of $i\beta_{0}$). Dashed (solid) curves correspond to the Fermi energy set to $\mu_{1}$ ($\mu_{2}$), with colors matching the corresponding band structures (and $\eta$) in [(a)]. Common parameters are $t=t_{z}=0.9$, $\lambda_{\rm so}=1$, and $\lambda_{\rm M}=0.8$.