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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 $P(K)\sim K^{-λ}$. It is well established that the model is characterized by classical mean-field exponents for $λ>5$. In this note we show that the specific-heat discontinuity $δc_h$ at the critical point remains $λ$-dependent even for $λ>5$: $δc_h=3(λ-5)(λ-1)/[2(λ-3)^2]$ and attains its mean-field value $δc_h=3/2$ only in the limit $λ\to \infty$. We compare this behaviour with recent measurements of the $d$ dependency of $δc_h$ made for the Ising model on lattices with $d>4$ [Lundow P.H., Markström K., Nucl. Phys. B, 2015, Vol. 895, 305].