content_image/339865.jpg content_image/339864.jpg content_image/339866.jpg content_image/339867.jpg content_image/339868.jpg content_image/339869.jpg content_image/339870.jpg
and 

<img src='content_image/339865.jpg'>

<img src='content_image/339864.jpg'>

<img src='content_image/339866.jpg'>

In these terms the statistical sum Z can be written as: 

<img src='content_image/339867.jpg'>

Because the matrix A$_{k}$ is symmetric and the vector ⃗ η$_{k}$ is real, we can use the expression for the last Gaussian integral: 

<img src='content_image/339868.jpg'>

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: 

<img src='content_image/339869.jpg'>

The corresponding total statistical sum is: 

This expression is used in Sec. IVC to calculate the specific heat. 

<img src='content_image/339870.jpg'>

## Appendix B: Angular degrees of freedom in order parameter fluctuations 

