 Solution

Work first on the infinite chain, or with a periodized exponential kernel. Put $\rho =e^{-\kappa }$ and use lattice Fourier momentum $q\in [-\pi ,\pi ]$. Summing the geometric series gives the exponentially coupled Ising inverse kernel:ThusAlthough the original kernel is long-ranged, its inverse contains only a constant and a cosine and therefore is nearest-neighbor. Fourier orthogonality givesSubstitution into part (a) yieldsThis exact bulk identity does not require replacing $\sinh \kappa$ by $\kappa$; small $\kappa$ is used later to motivate a smooth continuum field.

For a finite open chain the inverse has off-diagonal entries $-\rho /[J(1-{\rho }^2)]$, interior diagonal entries $(1+{\rho }^2)/[J(1-{\rho }^2)]$ and endpoint diagonal entries $1/[J(1-{\rho }^2)]$. Accordingly the displayed homogeneous potential acquires the endpoint correction $\rho (m_1^2+m_N^2)/[J(1+\rho )]$. These boundary terms do not change the bulk thermodynamic conclusion. Stating infinite-chain or periodized boundary conditions is necessary if the translation-invariant expression is asserted exactly.