The Ising model consists of $N$ particles, labelled by $i$, arranged on a $D$-dimensional Euclidean lattice with periodic boundary conditions. Each particle has spin up $s_{i}=+1$, or down $s_{i}=-1$, and the energy in the presence of a magnetic field $B$ is

$$E=-B \sum_{i} s_{i}-J \sum_{\langle i, j\rangle} s_{i} s_{j}$$

where $J>0$ is a constant and $\langle i, j\rangle$ indicates that the second sum is over each pair of nearest neighbours (every particle has $2 D$ nearest neighbours). Let $\beta=1 / k_{B} T$, where $T$ is the temperature.

(i) Express the average spin per particle, $m=\left(\sum_{i}\left\langle s_{i}\right\rangle\right) / N$, in terms of the canonical partition function $Z$.

(ii) Show that in the mean-field approximation

$$Z=C\left[Z_{1}\left(\beta B_{\mathrm{eff}}\right)\right]^{N}$$

where $Z_{1}$ is a single-particle partition function, $B_{\text {eff }}$ is an effective magnetic field which you should find in terms of $B, J, D$ and $m$, and $C$ is a prefactor which you should also evaluate.

(iii) Deduce an equation that determines $m$ for general values of $B, J$ and temperature $T$. Without attempting to solve for $m$ explicitly, discuss how the behaviour of the system depends on temperature when $B=0$, deriving an expression for the critical temperature $T_{c}$ and explaining its significance.

(iv) Comment briefly on whether the results obtained using the mean-field approximation for $B=0$ are consistent with an expression for the free energy of the form

$$F(m, T)=F_{0}(T)+\frac{a}{2}\left(T-T_{c}\right) m^{2}+\frac{b}{4} m^{4}$$

where $a$ and $b$ are positive constants.