content_image/342515.jpg content_image/342514.jpg content_image/342513.jpg content_image/342512.jpg content_image/342511.jpg
<img src='content_image/342515.jpg'>

We change the summation over k into integration and have: 

<img src='content_image/342514.jpg'>

After differentiating twice (43) (cf. Eq. (31)), we obtain the part δC$_{p}$ for the specific heat including Gaussian fluctuations. Finally, the total specific heat becomes 

<img src='content_image/342513.jpg'>

where we have added δC$_{p}$ to the mean-field part. Following our previous notation we set the integration limit as k$_{max}$ = ξ π $_{a}$, where we take the lattice parameter a = 9 . 5415 ˚ A [13]. In Fig. 10 we compare the theoretical results for the specific heat with the temperature dependence of Δ C$_{p}$ = C$_{p}$ |$_{BiMnO}$$_{3}$ − C$_{p}$ |$_{BiScO}$$_{3}$ . We see that the fluctuations overestimate the experimental behavior for T < T$_{RM}$ and underestimate the data for T > T$_{RM}$ . We attribute this (cf. Sec. III B) to the role of short-range order which gradually disappears as T increases above T$_{RM}$ . This is also the reason why the mean-field results match the experiment well for T < T$_{RM}$ . 

## D. Correlation lengths in multiferroics 

In an analogous manner one can calculate the evolution of the correlation length through the magnetic phase transition. 

The correlation function of an order parameter ψ ( ⃗ r ) in two distant points is defined as: 

<img src='content_image/342512.jpg'>

After a Fourier transform we can write g ( ⃗ r, ⃗ r $^{′}$) as: 

<img src='content_image/342511.jpg'>

