Topological spin multipolization and linear magnetoelectric coupling in two-dimensional antiferromagnets

TL;DR

The paper demonstrates a topological spin magnetoelectric effect in two-dimensional antiferromagnets, showing that a topological contribution to the spin magnetoelectric polarizability (SMP) arises when the electronic structure is captured by a massive 2D Dirac theory. Through two minimal lattice models—one with spin-orbit coupling and a Néel order, and another spin-orbit-free with double-Q order—it connects the SMP to the response of 2D Dirac semimetals and to a one-dimensional Z_2 crystalline topology, yielding a quasitopological electromagnetic response. A Landau-theory framework clarifies symmetry-allowed couplings, while continuum Dirac analyses reveal how Dirac mass and Zeeman-field–induced node shifts yield a sign-defined, topological SMP; 1D toy models reinforce the 1D origin of the effect. The results suggest concrete material realizations (e.g., CuMnAs) and point toward tunable magnetoelectric effects via Zeeman fields or engineered heterostructures, broadening the scope of topological magnetoelectric phenomena beyond 3D topological insulators.

Abstract

In this paper we predict that the magnetoelectric response of two-dimensional (2D) antiferromagnets is determined by the topology of the ground state. This topological magnetoelectric response, encoded in the spin magnetoelectric polarizability and its closely related spin multipolization, occurs when the electronic structure of the antiferromagnetic insulator is described by massive 2D Dirac fermions, and is therefore native to 2D, unlike the topological magnetoelectric effect of three-dimensional topological insulators. To demonstrate the topological contribution to the (spin) magnetoelectric polarizability, we compute the magnetoelectric polarizability microscopically for two distinct minimal lattice models: a spin-orbit coupled Néel antiferromagnet and a spin-orbit-free noncollinear antiferromagnet with double-$Q$ spin order. We show that the topological origin of the revealed magnetoelectric effect can be traced back to the electromagnetic response of topological semimetals in two dimensions, and hence is ultimately governed by a strong topological invariant in one dimension. Given this dimensional hierarchy, we further consider two minimal lattice models in one dimension, both one-dimensional variants of the 2D lattice models, and show that the magnetoelectric polarizability exhibits a clear signature of nontrivial crystalline topology. Possible material realizations are discussed.

Figures (11)

• Figure 1: (a) Schematic of the linear dispersion of a fourfold degenerate Dirac crossing at ${\bf q}=0$. The combination of inversion and time-reversal symmetry ensure a twofold degenerate spectrum away from the Dirac crossing. (b) Schematic of the splitting of the Dirac point into (2D) Weyl points in the presence of a Zeeman field $B$. The momentum separation $\delta q$ is proportional to $B$ and creates a region of effectively one-dimensional (1D) topologically nontrivial systems in the direction perpendicular to the separation.
• Figure 2: (a) Top panel: Sketch of buckled square lattice in two dimensions. The $A$ and $B$ (black and gray dots) sites a shifted in the negative and positive $z$ direction (i.e., out of the plane), respectively. Bottom panel: The spectrum of Eq. \ref{['eq:H_k-square']}, shown for $\lambda=0.6t_1$, featuring Dirac points at $M$, $X$, and $X'$ (see inset). (b) Top panel: Sketch of inversion-breaking Néel order in the buckled square lattice model. When the Néel vector points in the $z$ direction all three Dirac points are gapped (bottom panel).
• Figure 3: Splitting of the fourfold Dirac points in the presence of a Zeeman field. (a) A Zeeman field $B_x$ splits the Dirac points into pairs of twofold Dirac crossings along the $k_y$ axis. The gray shadings indicate regions where the Hamiltonian, when viewed as a collection of 1D models parametrized by $k_x$, has a nontrivial $\mathbb{Z}_2$ invariant with half-quantized polarization. (b) Similarly, a Zeeman field $B_y$ splits the Dirac points along the $k_x$ axis, creating a family of nontrivial 1D insulators along $k_x$.
• Figure 4: (a) Magnetoelectric polarizability $\alpha_{xx}=\alpha_{yy}$ of the buckled square model, calculated using Eq. \ref{['eq:SMP-4band']} and shown as a function of Néel order parameter $N_z$ (in units of $t_1$). Different curves correspond to $\lambda/t_1=0.4,0.6,0.8,1.0$, where larger values of $\lambda$ correspond to smaller (absolute) values of $\alpha_{xx}$. (b) Same as in (a) but only for $N_z/t_1>0$. The red dashed curves indicate analytical approximations obtained from considering only contributions from the continuum Dirac models at the high symmetry points.
• Figure 5: Magnetoelectric polarizabilities (a) $\alpha_{xx}$ and (b) $\alpha_{yy}$ of the buckled square lattice model when the symmetry breaking perturbation of Eq. \ref{['eq:delta-H_k-buckled']} is included. The symmetry breaking term gaps the Dirac point at $X'$, but preserves the Dirac points at $X$ and $M$, is indicated in the insets. In both panels the different curves correspond to $\lambda/t_1=0.4,0.6,0.8,1.0$, and we have set $\delta t =0.4t_1$. As in Fig. \ref{['fig:alpha_xx_buckled']}(b), the blue dotted curves are obtained numerically using Eq. \ref{['eq:SMP-4band']}, and the red dashed lines indicate analytical approximations obtained from a continuum Dirac model for the high symmetry points.