Consider a system of $N$ interacting spins. At low temperatures, the interactions ensure that all spins are either aligned or anti-aligned with the $z$ axis, even in the absence of an external field. At high temperatures, the interactions become less important and spins can point in either $\pm\hat{z}$ direction. If the heat capacity takes the form \[C_V = C_{\max}\!\left(\frac{2T}{T_0}-1\right) \quad\text{for } \frac{T_0}{2} < T < T_0 \quad\text{and}\quad C_V = 0 \text{ otherwise},\tag{1.1}\] determine $C_{\max}$.
Heat Capacity of an Interacting Spin System
Source: D. Tong, Statistical Physics, Problem Sheet 1, Question 4.