Solve exactly the problem of a field-free Ising chain with nearest-neighbor and next-nearest-neighbor interactions, so that

\[
H\left\{\sigma_{i}\right\}=-J_{1} \sum_{i} \sigma_{i} \sigma_{i+1}-J_{2} \sum_{i} \sigma_{i} \sigma_{i+2}
\]

and examine the various properties of interest of this model.

[Hint: Introduce a new variable \(\tau_{i}=\sigma_{i} \sigma_{i+1}= \pm 1\), with the result that

\[
H\left\{\tau_{i}\right\}=-J_{1} \sum_{i} \tau_{i}-J_{2} \sum_{i} \tau_{i} \tau_{i+1}
\]

which is formally similar to expression (13.2.1)].