Consider a double Ising chain such that the nearest-neighbor coupling constant along either chain is \(J_{1}\) while the one linking adjacent spins in the two chains is \(J_{2}\). Then, in the absence of the field,

\[
H\left\{\sigma_{i}, \sigma_{i}^{\prime}\right\}=-J_{1} \sum_{i}\left(\sigma_{i} \sigma_{i+1}+\sigma_{i}^{\prime} \sigma_{i+1}^{\prime}\right)-J_{2} \sum_{i} \sigma_{i} \sigma_{i}^{\prime}
\]

Show that the partition function of this system is given by

\[
\frac{1}{2 N} \ln Q \approx \frac{1}{2} \ln \left[2 \cosh K_{2}\left\{\cosh 2 K_{1}+\sqrt{ }\left(1+\sinh ^{2} 2 K_{1} \tanh ^{2} K_{2}\right)\right\}\right]
\]

where \(K_{1}=\beta J_{1}\) and \(K_{2}=\beta J_{2}\). Examine the various thermodynamic properties of this system.

Express the Hamiltonian \(H\) in a symmetric form by writing the last term as \(-\frac{1}{2} J_{2} \Sigma_{i}\left(\sigma_{i} \sigma_{i}^{\prime}+\sigma_{i+1} \sigma_{i+1}^{\prime}\right)\) and use the transfer matrix method.