June 26, 2002
We study first- and second-order phase transitions of ferromagnetic lattice models on scale-free networks, with a degree exponent $\gamma$. Using the example of the $q$-state Potts model we derive a general self-consistency relation within the frame of the Weiss molecular-field approximation, which presumably leads to exact critical singularities. Depending on the value of $\gamma$, we have found three different regimes of the phase diagram. As a general trend first-order transitions soften with decreasing $\gamma$ and the critical singularities at the second-order transitions are $\gamma$-dependent.
Similar papers 1
February 14, 2013
The Potts model is one of the most popular spin models of statistical physics. The prevailing majority of work done so far corresponds to the lattice version of the model. However, many natural or man-made systems are much better described by the topology of a network. We consider the q-state Potts model on an uncorrelated scale-free network for which the node-degree distribution manifests a power-law decay governed by the exponent \lambda. We work within the mean-field appro...
April 1, 2010
In this paper, we address the logarithmic corrections to the leading power laws that govern thermodynamic quantities as a second-order phase transition point is approached. For phase transitions of spin systems on d-dimensional lattices, such corrections appear at some marginal values of the order parameter or space dimension. We present new scaling relations for these exponents. We also consider a spin system on a scale-free network which exhibits logarithmic corrections due...
February 21, 2014
We study the critical behaviour of the $q$-state Potts model on an uncorrelated scale-free network having a power-law node degree distribution with a decay exponent $\lambda$. Previous data show that the phase diagram of the model in the $q,\lambda$ plane in the second order phase transition regime contains three regions, each being characterized by a different set of critical exponents. In this paper we complete these results by finding analytic expressions for the scaling f...
December 14, 2004
The Ising model in uncorrelated scale-free networks has been studied by means of Monte Carlo simulations. These networks are characterized by a degree (or connectivity) distribution $P(k) \sim k^{-\gamma}$. The ferromagnetic-paramagnetic transition temperature has been studied as a function of the parameter $\gamma$. For $\gamma > 3$ our results agree with earlier analytical calculations, which found a phase transition at a temperature $T_c(\gamma)$ in the thermodynamic limit...
November 25, 2022
Different models are proposed to understand magnetic phase transitions through the prism of competition between the energy and the entropy. One of such models is a $q$-state Potts model with invisible states. This model introduces $r$ invisible states such that if a spin lies in one of them, it does not interact with the rest states. We consider such a model using the mean field approximation on an annealed scale-free network where the probability of a randomly chosen vertex ...
April 12, 2007
We consider the ferromagnetic large-$q$ state Potts model in complex evolving networks, which is equivalent to an optimal cooperation problem, in which the agents try to optimize the total sum of pair cooperation benefits and the supports of independent projects. The agents are found to be typically of two kinds: a fraction of $m$ (being the magnetization of the Potts model) belongs to a large cooperating cluster, whereas the others are isolated one man's projects. It is show...
September 27, 2021
We consider a recently introduced generalization of the Ising model in which individual spin strength can vary. The model is intended for analysis of ordering in systems comprising agents which, although matching in their binarity (i.e., maintaining the iconic Ising features of `+' or `$-$', `up' or `down', `yes' or `no'), differ in their strength. To investigate the interplay between variable properties of nodes and interactions between them, we study the model on a complex ...
August 25, 2009
The antiferromagnetic Ising model in uncorrelated scale-free networks has been studied by means of Monte Carlo simulations. These networks are characterized by a connectivity (or degree) distribution P(k) ~ k^(- gamma). The disorder present in these complex networks frustrates the antiferromagnetic spin ordering, giving rise to a spin-glass (SG) phase at low temperature. The paramagnetic-SG transition temperature T_c has been studied as a function of the parameter gamma and t...
May 6, 1997
The effect of quenched impurities on systems which undergo first-order phase transitions is studied within the framework of the q-state Potts model. For large q a mapping to the random field Ising model is introduced which provides a simple physical explanation of the absence of any latent heat in 2D, and suggests that in higher dimensions such systems should exhibit a tricritical point with a correlation length exponent related to the exponents of the random field model by \...
June 7, 2006
We study a zero-temperature phase transition in the random field Ising model on scale-free networks with the degree exponent $\gamma$. Using an analytic mean-field theory, we find that the spins are always in the ordered phase for $\gamma<3$. On the other hand, the spins undergo a phase transition from an ordered phase to a disordered phase as the dispersion of the random fields increases for $\gamma > 3$. The phase transition may be either continuous or discontinuous dependi...