We change the summation over k into integration and have: 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 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$_{RM}$ . 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: After a Fourier transform we can write g ( ⃗ r, ⃗ r $^{′}$) as: 18 〈 δψ ( ⃗ r$_{1}$ ) δψ ( ⃗ r$_{2}$ ) 〉 = ∑ k$_{1}$,k$_{2}$ $^{〈}$ψ ∗ $_{k$_{1}$}$ψ$_{k}$$_{2}$ e i ( ⃗ k$_{2}$ − ⃗ k$_{1}$ ) · ⃗ $^{r$_{1}$}$ei ⃗ k$_{2}$ · ( ⃗ r$_{2}$ − ⃗ r$_{1}$ ) 〉 (46) g ( ⃗ r, ⃗ r $^{′}$) = $^{〈}$( ψ ( ⃗ r ) − ¯ ψ )( ψ ( ⃗ r $^{′}$) − ¯ ψ ) 〉 = $^{〈}$δψ ( ⃗ r ) δψ ( ⃗ r $^{′}$) $^{〉}$. (45) Δ C$_{p}$ = − T ∂ $^{2}$φ$_{0}$ ∂T 2 · a 2 $_{0}$Ω b + δC$_{p}$, (44) δF = − k$_{B}$T 2 ∫ k$_{max}$ 0 d $^{3}$k (2 π ) 3 ln$^{{}$ π 2 4 $^{(}$k$_{B}$ T C$_{m}$ ) 2 1 | ( A$_{m}$ + 1 $_{2}$a$_{3}$ k $^{2}$)( B$_{m}$ + 1 $_{2}$k$^{2}$) + 1 $_{4}$γ 2 $_{me}$|$^{}}$ . (43) δF = − k$_{B}$T ln Z (42) = − k$_{B}$T 2 ∑ k ln$^{{}$ π 2 4 $^{(}$k$_{B}$ T C$_{m}$ ) 2 1 | ( A$_{m}$ + 1 $_{2}$a$_{3}$ k $^{2}$)( B$_{m}$ + 1 $_{2}$k$^{2}$) + 1 $_{4}$γ 2 $_{me}$|$^{}}$ . D. Correlation lengths in multiferroics