FIG. 10: (Color online). Temperature dependence of the specific heat for BiMnO$_{3}$. Dotted line: the mean-field part of the specific heat; thick solid line: experimental data [13] was taken as C$_{p}$ |$_{BiM nO}$$_{3}$ − C$_{p}$ |$_{BiScO}$$_{3}$ . Thin solid line: specific heat after taking into account the Gaussian thermal fluctuations calculated for ξ = 2 . 71 · 10 − $^{8}$[ m ] and n = 1. and finally: where ⃗ r = ⃗ r$_{2}$ − ⃗ r$_{1}$ . To evaluate the coherence length for both the magnetic and the electric order parameters we need to calculate 〈 δm$_{-k}$δm$_{k}$ 〉 and 〈 δp$_{-k}$δp$_{k}$ 〉 . Using (A7) and (A8) we obtain the following expressions: The average values 〈 δm$_{-k}$δm$_{k}$ 〉 and 〈 δp$_{-k}$δp$_{k}$ 〉 can be calculated using the matrix A$_{k}$ defined in the Appendix A: 19 〈 δψ ( ⃗ r$_{1}$ ) δψ ( ⃗ r$_{2}$ ) 〉 = ∑ k 〈 δψ$_{−}$$_{k}$δψ$_{k}$ 〉 e i ⃗ k · ⃗ $^{r}$, (47) 〈 δm$_{-k}$δm$_{k}$ 〉 = $^{〈}$δm 2 1k + δm 2 2k $^{〉}$, (48) 〈 δp$_{-k}$δp$_{k}$ 〉 = $^{〈}$δp 2 1k + δp 2 2k $^{〉}$. (49) 1 2 〈 δp$_{-k}$δm$_{k}$ + δp$_{k}$δm$_{-k}$ 〉 = 〈 δp$_{1k}$m$_{1k}$ + δp$_{2k}$m$_{2k}$ 〉 . (50) 〈 δm$_{-k}$δm$_{k}$ 〉 = ∫ D η$_{k}$e − β $^{∑}$$_{k}$⃗ η T $_{k}$A$_{k}$ ⃗ η T k ( δm 2 1 k + δm 2 2 $_{k}$) ∫ D η$_{k}$e − β $^{∑}$$_{k}$⃗ η T $_{k}$A$_{k}$ ⃗ η T k , (51) 〈 δp$_{-k}$δp$_{k}$ 〉 = ∫ D η$_{k}$e − β $^{∑}$$_{k}$⃗ η T $_{k}$A$_{k}$ ⃗ η T k ( δp 2 1 k + δp 2 2 $_{k}$) ∫ D η$_{k}$e − β $^{∑}$$_{k}$⃗ η T $_{k}$A$_{k}$ ⃗ η T k . (52)