⃗ η$_{k}$ =      δp$_{1k}$ δp$_{2k}$ δm$_{1k}$ δm$_{2k}$$^{}$     , (A11) ⃗ η T k = ( δp$_{1k}$, δp$_{2k}$, δm$_{1k}$, δm$_{2k}$ ) , (A12) A$_{k}$ =      4 C$_{m}$ ( A$_{m}$ + 1 $_{2}$a$_{3}$ k $^{2}$) 0 2 C$_{m}$γ$_{me}$ 0 0 4 C$_{m}$ ( A$_{m}$ + 1 $_{2}$a$_{3}$ k $^{2}$) 0 2 C$_{m}$γ$_{me}$ 2 C$_{m}$γ$_{me}$ 0 4 C$_{m}$ ( B$_{m}$ + 1 $_{2}$k$^{2}$) 0 0 2 C$_{m}$γ$_{me}$ 0 4 C$_{m}$ ( B$_{m}$ + 1 $_{2}$k$^{2}$)$^{}$     . (A13) Z = ∫ D η$_{k}$e − β $^{∑}$$_{k}$⃗η T $_{k}$A$_{k}$ ⃗η T k ≡ ∏ k Z$_{k}$ . (A14) Z$_{k}$ = (2 π ) D 2 ( det A$_{k}$ ) − 1 2 , (A15) Z$_{k}$ = 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 $_{me}$| . (A16) Z = ∏ k ,k$_{z}$> 0 Z$_{k}$ (A17) 25 This expression is used in Sec. IVC to calculate the specific heat. The corresponding total statistical sum is: where D is the dimension of the vector ⃗ η$_{k}$ which in our case is equal to 4. The final expression for Z$_{k}$ after diagonalization of (A11) can be written as: Because the matrix A$_{k}$ is symmetric and the vector ⃗ η$_{k}$ is real, we can use the expression for the last Gaussian integral: In these terms the statistical sum Z can be written as: and Appendix B: Angular degrees of freedom in order parameter fluctuations