A magnetization m is the well-known hallmark of magnetic order in ferromagnets. A counterpart to m describing the magnetic order in antiferromagnets has conspicuously remained absent in the world of magnetism. We derive a Néel operator τ that describes collinear antiferromagnetic order, in the same way a magnetization m signals ferromagnetic order [1]. While m is even under space inversion and odd under time reversal, τ describes a toroidal moment that is odd under both symmetries. Thus m and τ quantify complementary aspects of magnetic order in solids. Orbital ferromagnetic order can be attributed to dipolar equilibrium currents that give rise to a magnetization. We show that, in the same way, antiferromagnetic order arises from quadrupolar equilibrium currents that generate the toroidal moment τ. Considering an antiferromagnetic diamond structure as an example (Fig. 1), we derive explicit analytical expressions for the Néel operator τ and the effects of AFM order on the itinerant electrons in this system [1]. As an example for the efficacy of our approach, we develop a unified theory for magnetoelectricity in quasi-2D paramagnets, ferromagnets and antiferromagnets. In magnetoelectric media, an electric field can induce a magnetization and a magnetic field can induce a polarization, while the system remains in thermal equilibrium [2,3]. This effect appeals to recent efforts to manipulate magnetic order electrically [4]. It requires that both space-inversion and time-reversal symmetry are broken. We present a comprehensive theory [1] for magnetoelectricity in quasi-2D systems, where antiferromagnetic order and toroidal moments play a central role. We obtain quantitative expressions for the magnetoelectric responses due to electric and magnetic fields that reveal explicitly, for the first time, the inherent duality of these responses required by thermodynamics.

![](https://assets.underline.io/uploads/markdown_image/1/image/fa70f074e078d7f874157bfe833bf03c.jpg)

Figure 1: The antiferromagnetic diamond structure breaks time-reversal symmetry θ and inversion symmetry I (though the joint operation θI remains a good symmetry).

Toroidal moments: hallmarks of collinear antiferromagnetic order

**Session chairs:** Jean Anne Incorvia, University of Texas at Austin and Vincent Baltz, Spintec



GOO - Antiferromagnetic Spintronics I

technical paper

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Roland Winkler

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Toroidal moments: hallmarks of collinear antiferromagnetic order

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