Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks.

Mariana Krasnytska (1) , Bertrand Berche (2) , Y. Holovatch (1) , R. Kenna (3)

1 ICMP - Institute for Condensed Matter Physics of NAS of Ukraine
2 Equipe 106, IJL - Groupe de Physique statistique = Statistical Physics Group [Institut Jean Lamour]
3 Applied Mathematics Research Centre

We analyse the partition function of the Ising model on graphs of two different types: complete graphs, wherein all nodes are mutually linked and annealed scale-free networks for which the degree distribution decays as P(k) ∼ k−λ. We are interested in zeros of the partition function in the cases of complex temperature or complex external field (Fisher and Lee–Yang zeros respectively). For the model on an annealed scale-free network, we find an integral representation for the partition function which, in the case λ > 5, reproduces the zeros for the Ising model on a complete graph. For 3 < λ < 5 we derive the λ-dependent angle at which the Fisher zeros impact onto the real temperature axis. This, in turn, gives access to the λ-dependent universal values of the critical exponents and critical amplitudes ratios. Our analysis of the Lee–Yang zeros reveals a difference in their behaviour for the Ising model on a complete graph and on an annealed scale-free network when 3 < λ < 5. Whereas in the former case the zeros are purely imaginary, they have a non zero real part in latter case, so that the celebrated Lee–Yang circle theorem is violated.

Domaines: Mécanique statistique [cond-mat.stat-mech]

Mariana Krasnytska, Bertrand Berche, Y. Holovatch, R. Kenna. Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks.. Journal of Physics A: Mathematical and Theoretical, 2016, 49 (13), ⟨10.1088/1751-8113/49/13/135001⟩. ⟨hal-01298246⟩