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arXiv:cond-mat/0702396

Ising model S=1, 3/2 and 2 on directed networks

F. W. S. Lima, Edina M. S. Luz

Published Feb 16, 2007• 0 citations
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Abstract

On directed} Barabasi-Albert and Small-World networks the Ising model with spin S=1, 3/2 and 2 is now studied through Monte Carlo simulations. In this model, the order-disorder phase transition of the order parameter is well defined on Small-World networks for Ising model with spin S=1. We calculate the value of the critical temperature T_c for several values of rewiring probability p of the directed Small-World network. This model on directed Small-World networks we obtained a second-order phase transition for p=0.2 and first-order phase transition for p=0.8. The critical exponentes beta/nu, gamma/nu and 1/νwere calculated for p=0.2. On directed Barabasi-Albert we show that no there is phase transition for Ising model with spin S=1, 3/2 and 2.

DOI
10.48550/arxiv.cond-mat/0702396

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On directed} Barabasi-Albert and Small-World networks the Ising model with spin S=1, 3/2 and 2 is now studied through Monte Carlo simulations. In this model, the order-disorder phase transition of the order parameter is well defined on Small-World networks for Ising model with spin S=1. We calculate the value of the critical temperature T_c for several values of rewiring probability p of the directed Small-World network. This model on directed Small-World networks we obtained a second-order phase transition for p=0.2 and first-order phase transition for p=0.8. The critical exponentes beta/nu, gamma/nu and 1/νwere calculated for p=0.2. On directed Barabasi-Albert we show that no there is phase transition for Ising model with spin S=1, 3/2 and 2.
In this work, we consider an Ising model which allows spin-spin interaction in the systems. We assume that two-level quantum systems are randomly located in N nodes of a complex annealed scale-free network described by the Barabasi-Albert model. It is defined by the power-law degree distribution of nodes. We consider the mean-field approach to the system described by the Ising Hamiltonian. At a certain level, the system is totally characterized by the order parameter Sz. It contains a critical inverse temperature β, which depends on parameter ζ2 as the ratio of the second to the first moment of the degree distribution. We have found that for ζ2, that exceeds its critical value ζ2,c, high temperature phase transition occurs that can be explained by the hubs and clusters which appear in scale-free networks.
Dynamical phase transition in the Ising model on a Barabási–Albert network under the influence of periodic magnetic field is studied using Monte-Carlo simulations. For a wide range of the system sizes N and the field frequencies, approximate phase borders between dynamically ordered and disordered phases are obtained on a plane h (field amplitude) versus T/Tc (temperature normalized to the static critical temperature without external field, Tc∝ ln N). On these borders, second- or first-order transitions occur, for parameter ranges separated by a tricritical point. For all frequencies of the magnetic field, position of the tricritical point is shifted toward higher values of T/Tc and lower values of h with increasing system size, i.e. the range of critical parameters corresponding to the first-order transition is broadened.
By using Monte Carlo simulation,we have studied phase transitions of the isotropic Ashkin-Teller model,for the ferromagnetic case,on a two-dimensional rewiring-type small world network,and obtained phase diagrams for several different values of the rewiring probability.It can be found that the rewiring probability have little effect on the structure of the phase diagrams,each of which consists of three phases,i.e.,the fully disordered paramagnetic phase, the partially ordered ferromagnetic phase and the fully ordered ferromagnetic phase,and there are first-order transitions between the fully ordered ferromagnetic phase and the fully disordered paramagnetic phase as well as between the fully ordered ferromagnetic phase and the partially ordered ferromagnetic phase.
In the recent study of the Ising model on a small world network by Pekalski [Phys. Rev. E 64, 057104 (2001)], a surprisingly small value of the critical exponent beta approximately 0.0001 has been obtained for the temperature dependence of the magnetization. We perform extensive Monte Carlo simulations of the same model and conclude, via the standard finite-size scaling of various quantities, that the phase transition in the model is of the mean-field nature, in contrast to the work by Pekalski, but in accordance with other existing studies.
The critical behavior of the Ising model on a one-dimensional network, which has long-range connections at distances l>1 with the probability theta(l) approximately l(-m), is studied by using Monte Carlo simulations. Through analyzing the Ising model on networks with different m values, this paper discusses the impact of the global correlation, which decays with the increase of m, on the phase transition of the Ising model. Adding the analysis of the finite-size scaling of the order parameter [M], it is observed that in the whole range of 0<m<2, a finite-temperature transition exists, and the critical exponents show consistence with mean-field values, which indicates a mean-field nature of the phase transition.