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as well [18] . As one can see, the system is characterized by two bare transition temperatures, T$_{f}$ and T$_{M}$ representing ferroelectric and ferromagnetic transitions, respectively. The form of the magnetoelectic coupling term ∝ γ ( PM ) 2 can be explained by using general symmetry arguments. Namely, the onset of ferroelectric order requires the breaking of spatial inversion symmetry, whereas the appearance of a spontaneous magnetization is connected with the breakdown of time reversal symmetry. The coupling term in (2) obeys those two conditions and allows for ferroic order with simultaneous nonzero P and M . 

In further calculations we use a dimensionless form of the free energy expansion (B5) which is obtained from (2) by dividing both sides of (2) by the constant value a 2 $_{0}$T 2 M b . In effect, we obtain: 

<img src='content_image/373250.jpg'>

where: Δ F = b a 2 $_{0}$T 2 M [ F ( P,M,T ) − F$_{0}$ ( T )], h = b 1 2 ( a$_{0}$T$_{M}$ ) 3 2 H$_{a}$ ≡ s$_{h}$H$_{a}$ , l = α 2 $_{0}$b a 2 $_{$_{0}$β}$, e = b a 2 $_{0}$T 2 M √ α$_{0}$T$_{f}$ β E$_{a}$ , ˜ M = M $_{m$_{0}$}$, m$_{0}$ = √ a$_{0}$T$_{M}$ b , ˜ P = P $_{p$_{0}$}$, p$_{0}$ = √ α$_{0}$T$_{f}$ β , γ$_{m}$ = γα$_{0}$ $_{βa$_{0}$}$. 

Within this formulation T$_{f}$ $_{T$_{M}$}$, l , and γ$_{m}$ are the material parameters. In this case, the bare magnetic transition temperature T$_{M}$ is assumed as always nonzero, whereas the ferroelectric temperature is T$_{f}$ ≥ 0. 

Taking derivatives of (B5) with respect to ˜ M and ˜ P , respectively, we obtain the following system of nonlinear equations: 

<img src='content_image/373249.jpg'>

<img src='content_image/373248.jpg'>

In the limit of zero electric field, we obtain the following relation between ˜ P and ˜ M from (4): 

<img src='content_image/373247.jpg'>

