TY - JOUR

T1 - Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks

AU - Krasnytska, M.

AU - Berche, B.

AU - Holovatch, Y.

AU - Kenna, Ralph

PY - 2016/4/1

Y1 - 2016/4/1

N2 - We analyze the partition function of the Ising model on graphs of twodifferent types: complete graphs, wherein all nodes are mutually linked and annealedscale-free networks for which the degree distribution decays as P(k) ∼ k−λ. We areinterested in zeros of the partition function in the cases of complex temperature orcomplex external field (Fisher and Lee-Yang zeros respectively). For the model on anannealed scale-free network, we find an integral representation for the partition functionwhich, in the case λ > 5, reproduces the zeros for the Ising model on a completegraph. For 3 <λ <5 we derive the λ-dependent angle at which the Fisher zerosimpact onto the real temperature axis. This, in turn, gives access to the λ-dependentuniversal values of the critical exponents and critical amplitudes ratios. Our analysisof the Lee-Yang zeros reveals a difference in their behaviour for the Ising model on acomplete graph and on an annealed scale-free network when 3 <λ <5. Whereas inthe former case the zeros are purely imaginary, they have a non zero real part in lattercase, so that the celebrated Lee-Yang circle theorem is violated.

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

U2 - 10.1088/1751-8113/49/13/135001

DO - 10.1088/1751-8113/49/13/135001

M3 - Article

SN - 1361-6447

SN - 1751-8121

VL - 49

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

IS - 13

M1 - 135001

ER -