Micorscopic theory of temperature-dependent magnetoelectric effect in Cr$_{2}$O$_{3}$

We study the temperature dependence of the magnetoelectric effect in Cr$_{2}$O$_{3}$ by considering the coupling between electric polarization and spins induced by Heisenberg exchange interactions. The form of the coupling is obtained by symmetry analysis and its strength is calculated by {\it ab initio} methods. Using Monte Carlo simulations, we evaluate the temperature dependence of the largest component of the magnetoelectric susceptibility. The quantitative agreement of our results with experimental measurements shows that the dominant contribution to the linear magnetoelectric effect originates from nonrelativistic exchange interactions and spin fluctuations. The approach used can be applied to study other magnetoelectrics with collinear spin ordering and opens a new route for the design of materials with large magnetoelectric effect at high temperatures.