We show that nonrelativistic exchange interactions and spin fluctuations can give rise to a linear magnetoelectric effect in collinear antiferromagnets at elevated temperatures that can exceed relativistic magnetoelectric responses by more than 1 order of magnitude. We show how symmetry arguments, ab initio methods, and Monte Carlo simulations can be combined to calculate temperature-dependent magnetoelectric susceptibilities entirely from first principles. Introduction.-Recent years have seen a resurgence of interest in materials with coupled electric and magnetic dipoles motivated by the prospect of controlling spins with applied voltages and charges with applied magnetic fields in novel multifunctional devices. The simplest form of such a control is the linear magnetoelectric (ME) coupling between electric polarization and an applied magnetic field or, conversely, between magnetization and an applied electric field. Although the linear ME effect was theoretically predicted and experimentally discovered more than 50 years ago [1], finding technologically useful materials displaying strong ME coupling at room temperature remains a challenging problem [2].Recent progress in the related field of multiferroic materials, in which ferroelectric polarizations are induced by noncentrosymmetric magnetic orderings, has led to a clarification of the microscopic origins for ME coupling [3]. In particular, two distinct coupling mechanisms have been identified. The first arises from relativistic effects linking electron spin and orbital momentum, resulting in the antisymmetric S 1 Â S 2 interaction between spins of different magnetic ions. The strength of this Dzyaloshinksii-Moriya interaction depends on polar displacements of ions, which can make magnets with noncollinear spiral orders becoming ferroelectric [4][5][6][7]. In the second mechanism, polar deformations of the lattice are induced by Heisenberg spin exchange interactions S 1 Á S 2 , originating from the Fermi statistics of electrons [8,9]. This nonrelativistic mechanism can give rise to stronger spin-lattice couplings than those resulting from relativistic effects, which tend to be relatively weak in 3D transition metal compounds. Indeed, in multiferroics, the electric polarizations induced by exchange interactions in Y 1Àx Lu x MnO 3 and GdFeO 3 exceed the largest polarizations observed in spiral multiferroics by 1 order of magnitude [10,11]. It was recently suggested that Heisenberg exchange can also give rise to a relatively strong linear ME effect [12] which, however, seemed to require rather special noncollinear spin orderings and crystal structures, making it difficult to find such materials in nature.