Disentangling Anomalous Hall Effect Mechanisms and Extra Symmetry Protection in Altermagnetic Systems

Abstract

We investigate the evolution of Anomalous Hall Conductivity (AHC) in a coplanar and collinear antiferromagnetic system with varying spin canting angles. A tight-binding model based on three t2g-orbitals in a body-centered tetragonal lattice is constructed, where the inclusion of third-nearest neighbor hopping is demonstrated to be essential for capturing the characteristic energy band splitting of altermagnetic materials. By employing a symmetry analysis based on spin space groups and treating spin-orbit coupling (SOC) as a perturbation, we theoretically distinguish and numerically verify two origins of the transverse transport: the conventional anomalous Hall effect (AHE) induced by net magnetization and the Crystal Hall Effect (CHE) arising from specific crystal symmetries. Our results show that the conductivity components driven by these two mechanisms follow distinct trigonometric dependencies on the canting angle. Crucially, we identify a hidden C110 rotational symmetry that has been previously overlooked in static magnetic group analyses. By expanding the AHC in terms of spin orientation vectors, we demonstrate that this symmetry acts as a bridge connecting distinct magnetic configurations with different canting angles, thereby strictly protecting the equivalence of orthogonal conductivity components in the collinear system.

Figures (6)

• Figure 1: Schematic illustration of the body-centered tetragonal lattice structure adopted in the tight-binding model. The grey and red spheres represent the transition metal (magnetic) and ligand ions, respectively. The colored arrows indicate the hopping integrals considered in the Hamiltonian: $t_1$ (cyan) corresponds to the nearest-neighbor hopping between sublattices, while $t_2$ (green) and $t_3$ (magenta) denote the next-nearest and third-nearest neighbor hoppings, respectively. The vectors $t_{shell1}$ (blue) and $t_{shell2}$ (red) illustrate the anisotropic hopping paths that distinguish the local environments of the two sublattices, which is essential for capturing the altermagnetic symmetry.
• Figure 2: Energy dispersion along high symmetry line in tetragonal lattice with SOC, which contributes to the mix of the two settings of magnetization. The canting angle is set to 0. These two are without (a) and with (b) 3rd hopping.
• Figure 3: This figure presents a top-down view along the $z$-axis. The magnetic moments at the body-centered and corner sites are symmetric with respect to the $C_{2y}^{s}$ rotation axis. The vectors ${l}_2$ and ${l}_3$ lie within the plane, while ${l}_1$ aligns with the $z$-axis, perpendicular to the plane.
• Figure 4: Calculated AHC as a function of the spin orientation angle $\eta$. (a) The disentanglement of AHC mechanisms in the coplanar antiferromagnetic system (Fermi energy $\mu/t = -1.25$). For visual clarity and consistency in subsequent processing, the AHC values are treated as their negatives throughout. The red circles and black squares represent the conventional magnetization-induced AHE and the crystal-symmetry-induced CHE, respectively. The solid lines indicate the theoretical trigonometric fits expanded up to the 5th order, confirming their distinct symmetry origins. The comprehensive multipole expansion basis and corresponding fitting parameters are detailed in Table\ref{['tab:hall_fitting']}. (b) The evolution of orthogonal AHC components ($\sigma_x$ and $\sigma_y$) in the collinear system (Fermi energy $\mu/t = 1$). The solid curves show the fitted results ($C=D=0.0086$). Note that the strict equivalence of $\sigma_x$ and $\sigma_y$ at specific angles is protected by the hidden $C_{110}$ rotational symmetry.
• Figure 5: Calculated spin-polarized electronic band structure of $\mathrm{NiF_{2}}$ with SOC. The bands exhibit characteristic momentum-dependent spin splitting along the high-symmetry paths $M-\Gamma$ and $A-Z$, serving as a hallmark of $d$-wave altermagnetism. In contrast, the bands remain strictly spin-degenerate along the other lines due to symmetry protection.