Anomalous Hall effect in metallic collinear antiferromagnets

TL;DR

The paper develops minimal 2D models of metallic, collinear Néel-ordered antiferromagnets to explain the anomalous Hall effect (AHE) in systems with vanishing net moment. By combining symmetry analysis (Dzyaloshinskii's invariants) with explicit Berry-curvature calculations, it shows that AHE can arise only for specific Néel-vector directions in ferrimagnets and weak ferromagnets, where momentum-dependent exchange and spin-orbit coupling cooperate to lift degeneracies. The work provides concrete Hamiltonians and derived Berry-curvature expressions that validate the invariants and highlight why the AHE signal is typically small due to opposing contributions from conduction bands. These results clarify the role of symmetry in enabling AHE in AFMs and connect to broader concepts such as altermagnetism and spin-splitting in antiferromagnetic metals.

Abstract

We propose and theoretically study minimal models of Néel ordered collinear antiferromagnets exhibiting the anomalous Hall effect. For simplicity, we consider two-dimensional models of antiferromagnets with two magnetic sublattices on a square lattice. We provide explicit examples of a Néel ordered ferrimagnet and a Dzyaloshinskii weak ferromagnet. We analyze Dzyaloshinskii's invariants for the existence of spontaneous magnetization in these Néel ordered systems. As a result, we find that the anomalous Hall effect is allowed only for specific directions of the Néel order, dictated by the crystal lattice symmetries. Microscopic calculations of the Berry curvature for the studied systems confirm the validity of these Dzyaloshinskii's invariants. We show that the anomalous Hall effect mechanism in these antiferromagnets arises from the interplay of momentum-dependent exchange interaction of conducting fermions with the Néel order and the spin-orbit coupling, both originating from the broken symmetries that permit the Dzyaloshinskii's invariant in the system.

Figures (6)

• Figure 1: Left: An example of a genuine $d$-wave Néel ordered antiferromagnet. The Néel order is given by the sites with $\pm \mathbf{m}$. The green atom is non-magnetic and can be positioned either in the plane of the lattice or lifted from it. For simplicity, we assume no fermion tunneling through the green atom. Right: Contour plot of the Fermi surfaces of conducting fermions described by Hamiltonian Eq. (\ref{['genuine']}) for $m=2\xi$ and $t=0.15\xi$, and $\mu = 2.2\xi$ (in units of $\xi$). The Néel order is in the $z$-direction. The red plot is for spin-up fermions, while the blue is for spin-down.
• Figure 2: (a): Lattice of a Néel ordered ferrimagnet. (b) Description of the spin-orbit coupling. A square is an atom which is either on the bottom (cyan color) in $z-$direction or on top (purple color) of the link (as shown in (c)). Cyan/purple arrow is the direction of the spin-orbit coupling for a direction of fermion hopping defined by the black arrow. If the direction of the black arrow changes sign, the direction of the cyan/purple arrow will do so as well. There are two Dzyaloshinskii's invariants in the system: $M_{z}L_{z}$ and $M_{z}(L_{x}-L_{y})$.
• Figure 3: Plot of the anomalous Hall conductivity as a function of temperature. Left: ferrimagnetic model Eq. (\ref{['ferri']}) for $m_{x}=m_{y}=0$. Right: weak ferromagnet model Eq. (\ref{['weakMS']}) for $m_{x}=m_{z}=0$. In both plots $m=2\xi$, $t=0.15\xi$, $\gamma=\eta_{1}=\eta_{2}=0.1\xi$, and Fermi level was chosen $\mu = 2\xi$. It was assumed that $h=2\pi \hbar \equiv 1$.
• Figure 4: (a) A model of mirror-symmetric weak ferromagnet. A combination of a mirror reflection in the $x-z$ plane (crossing the center of the vertical link) and time-reversal operations is the symmetry which connects the two sublattices. (b) The green atom is lifted from the plane of the lattice. This is needed to eliminate a combination of reflection in the $x-y$ plane and time-reversal from the symmetries of the system. As a result, spin-orbit coupling acquires in-plane components shown by the green arrows. Thus, Dzyaloshinskii's invariant of weak Dzyaloshinskii's ferromagnetism is $M_{z}L_{y}$ in this model.
• Figure 5: Model of a ferrimagnet. Néel order is given by red and blue sites. Parameters $\xi$, $d$, $t$, and $\eta$ stand for various fermion tunneling processes described in the tight-binding model Eq. (\ref{['ferriA']}). Black arrows are the directions of fermion tunnelings corresponding to $+$ sign of the spin-orbit coupling.