Theory of magnetic field-induced metaelectric critical end point in BiMn$_2$O$_5$

A recent experiment on the multiferroic BiMn$_2$O$_5$ compound under a strong applied magnetic field revealed a rich phase diagram driven by the coupling of magnetic and charge (dipolar) degrees of freedom. Based on the exchange-striction mechanism, we propose here a theoretical model with the intent to capture the interplay of the spin and dipolar moments in the presence of a magnetic field in BiMn$_2$O$_5$. Experimentally observed behavior of the dielectric constants, magnetic susceptibility, and the polarization is, for the most part, reproduced by our model. The critical behavior observed near the polarization reversal $(P=0)$ point in the phase diagram is interpreted as arising from the proximity to the critical end point.