Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 72 / 4 / a / Solution

Write the finite-chain spin vector as $\sigma$ and the exchange matrix as $\mathsf{J}$. For $J>0$ and $\kappa >0$ this matrix is positive definite, as can be seen from the positive Fourier symbol in part (b), or from the explicit positive open-chain inverse. The linear-source multivariate Gaussian integral, obtained by completing the square, givesIndeed the exponent is $-(m-\mathsf{J}\sigma {)}^T{\mathsf{J}}^{-1}(m-\mathsf{J}\sigma )+{\sigma }^T\mathsf{J}\sigma$. Insert this Hubbard–Stratonovich transformation into the spin partition function. The field term remains $h{\sum }_i{\sigma }_i$, and each independent spin sum isConsequentlyThe interaction convention counts the full ordered pair sum, with no factor of one-half, so the auxiliary coupling is $2m_i{\sigma }_i$. Diagonal terms $J_{ii}{\sigma }_i^2$ only contribute a constant to the spin Hamiltonian; retaining them makes the given positive exchange matrix and normalization convenient. The auxiliary field is not itself the physical spin magnetization: the conditional spin expectation is $\tanh (2m_i+h)$, so $⟨{\sigma }_i⟩=⟨\tanh (2m_i+h)⟩$ in the auxiliary-field integral.