On the discontinuity of the specific heat of the Ising model on a scale-free network

We consider the Ising model on an annealed scale-free network with node-degree distribution characterized by a power-law decay $P(K)\sim K^{-\lambda}$. It is well established that the model is characterized by classical mean-field exponents for $\lambda>5$. In this note we show that the specific-heat discontinuity $\delta c_h$ at the critical point remains $\lambda$-dependent even for $\lambda>5$: $\delta c_h=3(\lambda-5)(\lambda-1)/[2(\lambda-3)^2]$ and attains its mean-field value $\delta c_h=3/2$ only in the limit $\lambda\to \infty$. We compare this behaviour with recent measurements of the $d$ dependency of $\delta c_h$ made for the Ising model on lattices with $d>4$ [Lundow P.H., Markstr\"{o}m K., Nucl. Phys. B, 2015, Vol. 895, 305].