Abstract
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 network where both the spin strength and degree distributions are governed by power laws. We show that in the annealed network approximation, thermodynamic functions of the model are self-averaging and we obtain an exact solution for the partition function. This allows us derive the leading temperature and field dependencies of thermodynamic functions, their critical behavior, and logarithmic corrections at the interface of different phases. We find the delicate interplay of the two power laws leads to new universality classes.
Highlights
It is almost futile, and perhaps impossible, to comprehensively list the advances in understanding of various phenomena in physics and beyond that were achieved due to the Ising model
Considering the critical behavior of a spin system on a complex network, special attention has been paid to scale-free networks, which are characterized by a power-law decay of a node degree distribution function: p ( K ) = cλ K −λ, Kmin ≤ K ≤ Kmax, (2)
It is well established that even a weak dilution by non-magnetic components may lead to crucial changes in the behavior of magnetically ordered systems. If such a dilution is implemented in a quenched fashion, changes in the universality class of the Ising model [34] are governed by the Harris criterion [63]
Summary
Perhaps impossible, to comprehensively list the advances in understanding of various phenomena in physics and beyond that were achieved due to the Ising model. An example is the q-state Potts model [21,22] which keeps the discrete symmetry of the Ising model, generalizing it from Z2 to Zq. As a result, each agent (spin) can take on only a finite number of states, the binarity is lost for any q 6= 2. In a recent short communication [26], we reported on the peculiarities of the generalized Ising model when the random spin length is governed by a power-law decaying distribution function. The Potts model, on the other hand, maintains discrete variables, but relaxes the number of single-site spin states We consider another generalization of the Ising model. Considering the critical behavior of a spin system on a complex network, special attention has been paid to scale-free networks, which are characterized by a power-law decay of a node degree distribution function:. We first formulate the annealed network approximation we will be dealing with
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