Consider a crystalline lattice with Ising spins $s_{\ell}=\boldsymbol{\ell}$ at each site $\boldsymbol{\ell}$. In the presence of an external field $\mathbf{H}=\left(0,0, H_0\right)$, the Hamiltonian of the system may be written as
$$
H=-J \sum_{\ell \in} s_{\boldsymbol{e}} s_{\boldsymbol{e}}-\mu_0 H_0 \sum_{\boldsymbol{\ell}} s_{\boldsymbol{\ell}},
$$
where $J>0$ is a constant and the sum $\sum_{\text {CE }}$ is over all nearest-neighbor sites only (each site has $p$ nearest neighbors).
(a) Write an expression for the free energy of the system at temperature $T$ (do not try to evaluate it).
(b) Using the mean-field approximation, derive an equation for the spontaneous magnetization $m=\left\langle s_0\right\rangle$ for $\mathbf{H}_0=0$ and calculate the critical temperature $T_{\mathrm{c}}$ below which $m \neq 0$.
(c) Calculate the critical exponent $\beta$ defined by $m\left(T, \mathbf{H}_0=0\right) \sim$ const. $\left(1-T / T_c\right)^\beta$ as $T \rightarrow T_c$.
(d) Describe the behavior of the specific heat at constant $\mathbf{H}_0, C\left(\mathbf{H}_0=\right.$ $0)$, near $T=T_c$.