Ferromagnet with a noncollinear antiferromagnetic order and anomalous Hall effect

TL;DR

The paper investigates anomalous Hall behavior in a metallic system where a noncollinear antiferromagnetic order interacts with conducting fermions via indirect tunneling through localized spins, yielding a momentum-dependent exchange term that is odd in momentum and resembles Rashba spin–orbit coupling but breaks time-reversal symmetry. In the ferromagnetic case, this mechanism produces a Berry-curvature–driven anomalous Hall effect without intrinsic SOC, accompanied by chiral edge states in the bulk gap. The work provides an analytic framework for AHE arising from noncollinear AF order coexisting with ferromagnetism and may be relevant to altermagnets and related systems. The results emphasize symmetry breaking and tunneling processes as alternative routes to spin–orbit–induced transport phenomena in complex magnetic textures.

Abstract

In this paper we introduce a theoretical model of a metallic magnetic system with noncollinear antiferromagnetic order. We introduce a mechanism of indirect interaction of conducting fermions with localized spins based on the tunneling processes of conducting fermions through the localized spins. We demonstrate that interaction of conducting fermions with noncollinear antiferromagnetic order results in odd in momentum spin-momentum locking. The interaction resembles Rashba spin-orbit coupling but breaks the time-reversal symmetry. As a result, we show that a ferromagnet with the noncollinear antiferromagnetic order is an insulator with anomalous Hall effect occurring without any spin-orbit coupling.

Figures (6)

• Figure 1: (a): a model of a noncollinear antiferromagnetic order. Black circles are the sites conducting fermions reside on. Red circles are sites with a localized spin. Direction of the spin on the red site is set by the red arrow. There is a tunneling of conducting fermions between the black sites. Fermion energy at the red site is higher than the one on the black. Thus, there is a tunneling of conducting fermions through the red sites which results in the noncollinear antiferromagnetic order interaction given in Eq. \ref{['antitoroidal']}. (b): a model of three sites on a line used as an example where fermion's momentum-dependent exchange interaction with localized spin appears.
• Figure 2: Two different models of genuine Néel ordered antiferromagnets on a checkerboard lattice. Red and blue sites correspond to localized fermions which form the Néel order (in $z-$direction, for example), in which spins on the red/blue sites point in up/down in $z-$direction. Conducting fermions in the model (a) reside on the colored sites, while in the model (b) they reside on black sites. The green atom shown in (a) is non-magnetic and it is assumed it blocks fermion tunneling through it. Hence, there is an anisotropic tunneling processes of conducting fermions described by the dashed lines in the model (a) which result in the $d-$wave spin-splitting described in Eq. (\ref{['checker2']}). In the model (b) the dashed lines describe Heisenberg exchange interaction between localized spins. Direct conducting fermion tunneling in the model (b) is between black sites, while there are also two-step tunneling processes through the colored sites in correspondence with Fig. (\ref{['fig:fig1']}b). The latter processes result in the $d-$wave spin-splitting of conducting fermions described in Eq. (\ref{['dwave']}).
• Figure 3: Spectrum of the system with noncollinear antiferromagnetic order described by Hamiltonian $\hat{H} = \hat{H}_{\mathrm{nc}}+\hat{H}_{\mathrm{fm}}$ defined in Eq. (\ref{['antitoroidal']}) and Eq. (\ref{['fm']}). Left plot is for $h_{z} = 0$ in which the conduction and valence bands touch at four points. Right: finite magnetic field, $h_{z} = 0.5\xi$, opens up a gap at these points. Both plots are for $\gamma =\xi$ in order to highlight gap opening. Physical prameters $h_{z}$ and $\gamma$ are in units of $\xi$, such that $\xi=1$.
• Figure 4: Anomalous Hall effect conductivity $\sigma_{\mathrm{AHE}}$ plotted as a function of $h_{z}$ at $T=0$. Blue: $\gamma =0.1 \xi$, yellow $\gamma=0.3 \xi$, and green $\gamma=0.6 \xi$. It was assumed that $h=2\pi \hbar \equiv 1$.
• Figure 5: Spectrum of fermions numerically calculated in a strip geometry. Parameters are chosen to be $\xi = \gamma = h_{z} = 1$ for simplicty and in order to highlight the edge states in the gap. The edge states are there as long as $h_{z}\neq 0$ and $\vert h_{z}\vert<2\xi$.