Consider a spherical model whose spins interact through a long-range potential varying as \((a / r)^{d+\sigma}(\sigma>0), r\) being the distance between two spins. This replaces the quantity \(\left(\lambda-\mu_{\mathbf{k}}\right)\) of equations (13.5.16) and (13.5.57) by an expression approximating \(J\left[\phi+\frac{1}{2}(k a)^{\sigma}\right]\) for \(\sigma<2\) and \(J\left[\phi+\frac{1}{2}(k a)^{2}\right]\) for \(\sigma>2\); note that the nearest-neighbor interaction corresponds to the limit \(\sigma \rightarrow \infty\) and hence to the latter case.

Assuming \(\sigma\) to be less than 2, show that the above system undergoes a phase transition at a finite temperature \(T_{c}\) for all \(d>\sigma\). Further show that the critical exponents for this model are

\[
\begin{gathered}
\alpha=\frac{d-2 \sigma}{d-\sigma}, \quad \beta=\frac{1}{2}, \quad \gamma=\frac{\sigma}{d-\sigma}, \quad \delta=\frac{d+\sigma}{d-\sigma} \\
u=\frac{1}{d-\sigma}, \quad \eta=2-\sigma
\end{gathered}
\]

for \(\sigma
\[
\alpha=0, \quad \beta=\frac{1}{2}, \quad \gamma=1, \quad \delta=3, \quad u=\frac{1}{\sigma}, \quad \eta=2-\sigma
\]

for \(d>2 \sigma\).