Microscopic theory on the electromagnetic phase diagram for multiferroic RMnO$_3$: A-type, incommensurate spiral, and E-type spin phases

We have theoretically studied the electromagnetic phase diagram for the multiferroic Mn perovskites ($R$MnO$_3$), which exhibits successive three magnetic phases of A-type, incommensurate spiral, and E- type spin orderings at low temperatures. We first show that bond alternations in the $J_1$-$J_2$ model successfully reproduce successive transitions among above three phases. We then studied the diagram by using a realistic spin model of the Mn perovskites, which contains the spin exchanges coupled with phonons, the DM interactions, and the single-ion spin anisotropies. This realistic model enables us to solve many puzzling issues of the diagram such as spin spiral-plane flop between $ab$ to $bc$, anomalous temperature dependence of electric polarizations, and ferroelectricity in the E-type phase. Comparison between above two studies highlights the roles of each interaction and anisotropy on the related intriguing phenomena.