Multiferroic Thermodynamics

We have studied the thermodynamic properties of a multiferroic that couples ferromagnetic and ferroelectric order. Some of the results are independent of the form of the free energy. We calculate the temperature dependence of the electric, magnetic, and magnetoelectric susceptibilities. The cross susceptibility has a temperature dependence related to the mixed (with respect to E and B) derivatives of the specific heat. The phase transitions are all second order. In particular, the phase boundary T$_M$(E), where T$_M$ is the lower magnetic transition as a function of electric field, is described by the Ehrenfest relation. The magnetoelectric susceptibility is nonzero only below the lower of the two transition temperatures. We study the properties of the specific heat, with and without the inclusion of gaussian fluctuations. The perturbative renormalization group is used to understand the fixed points of the theory, and we include a discussion of the effect inhomogeneities have for this model.