The one-dimensional Ising model consists of a set of $N$ spins $s_{i}$ with Hamiltonian

$$H=-J \sum_{i=1}^{N} s_{i} s_{i+1}-\frac{B}{2} \sum_{i=1}^{N}\left(s_{i}+s_{i+1}\right)$$

where periodic boundary conditions are imposed so $s_{N+1}=s_{1}$. Here $J$ is a positive coupling constant and $B$ is an external magnetic field. Define a $2 \times 2$ matrix $M$ with elements

$$M_{s t}=\exp \left[\beta J s t+\frac{\beta B}{2}(s+t)\right]$$

where indices $s, t$ take values $\pm 1$ and $\beta=(k T)^{-1}$ with $k$ Boltzmann's constant and $T$ temperature.

(a) Prove that the partition function of the Ising model can be written as

$$Z=\operatorname{Tr}\left(M^{N}\right)$$

Calculate the eigenvalues of $M$ and hence determine the free energy in the thermodynamic limit $N \rightarrow \infty$. Explain why the Ising model does not exhibit a phase transition in one dimension.

(b) Consider the case of zero magnetic field $B=0$. The correlation function $\left\langle s_{i} s_{j}\right\rangle$ is defined by

$$\left\langle s_{i} s_{j}\right\rangle=\frac{1}{Z} \sum_{\left\{s_{k}\right\}} s_{i} s_{j} e^{-\beta H}$$

(i) Show that, for $i>1$,

$$\left\langle s_{1} s_{i}\right\rangle=\frac{1}{Z} \sum_{s, t} s t\left(M^{i-1}\right)_{s t}\left(M^{N-i+1}\right)_{t s}$$

(ii) By diagonalizing $M$, or otherwise, calculate $M^{p}$ for any positive integer $p$. Hence show that

$$\left\langle s_{1} s_{i}\right\rangle=\frac{\tanh ^{i-1}(\beta J)+\tanh ^{N-i+1}(\beta J)}{1+\tanh ^{N}(\beta J)}$$