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International Journal of Modern Physics B

DYNAMICAL PHASE TRANSITION IN THE ISING MODEL ON A SCALE-FREE NETWORK

A. Krawiecki

Published Dec 30, 2005• 13 citations
Publisher

Abstract

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.

DOI
10.1142/s0217979205033017

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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.
The dynamic phase transition has been studied in the three-dimensional kinetic Ising model in the presence of a time-varying(sinusoidal)magnetic field by Monte Carlo simulation.The nature of the transition is characterized by studying the distribution of the order parameter.The system is in contact with an isothermal heat bath at temperature T.We varied the magnitude of the magnetic field,keeping the frequency of the magnetic field unchanged.For the lower values of the environment temperature the transition is discontinuous and it is continuous for higher values of the environment temperature,indicating the existence of a tricritical point (TCP) on the phase boundary.The TCP shifts towards higher temperature region with the decrease of frequency.i.e:TTCP=1.33×e(-ω/30.7)
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.
We have studied the nonequilibrium dynamic phase transitions of both three-dimensional (3D) kinetic Ising and Heisenberg spin systems in the presence of a perturbative magnetic field by Monte Carlo simulation. The feature of the phase transition is characterized by studying the distribution of the dynamical order parameter. In the case of anisotropic Ising spin system (ISS), the dynamic transition is discontinuous and continuous under low and high temperatures respectively, which indicates the existence of a tri-critical point (TCP) on the phase \linebreak boundary separating low-temperature order phase and high-temperature disorder phase. The TCP shifts towards the higher temperature region with the decrease of frequency, i.e. T TCP =1.33 × exp(−ω/30.7). In the case of the isotropic Heisenberg spin system (HSS), however, the situation on dynamic phase transition of HSS is quite different from that of ISS in that no stable dynamical phase transition was observed in kinetic HSS after a threshold time. The evolution of magnetization in the HSS driven by a symmetrical external field after a certain duration always tends asymptotically to a disorder state no matter what an initial state the system starts with. The threshold time τ depends upon the amplitude H 0 , reduced temperature T / T C and the frequency ω as τ = C .ω α . H 0 −β .( T / T C ) −γ .
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.
The dynamical phase transitions of the kinetic Ising model in the presence of a random magnetic field with a bimodal probability distribution is studied by using effective-field theory (EFT) with correlations. We have used a Glauber-type stochastic dynamic to describe the time evolution of the system, where the system strongly depends on the H≡√<H(i)(2)>(c) root mean square deviation of the magnetic field. The EFT dynamic equation is given for the simple cubic lattice (z=6), and the dynamic order parameter is calculated. The system presents ferromagnetic and paramagnetic states for low and high temperatures, respectively. Our results predict first-order transitions at low temperatures and large disorder strengths, which corresponds to the existence of a nonequilibrium tricritical point (TCP) in a phase diagram in the T-H plane. We compare the results with the equilibrium phase diagram, where only the first-order line is different. Our qualitative results are compatible with recent Monte Carlo simulations.
A Monte Carlo method has been used to study a simple cubic Ising ferromagnet in a random quenched magnetic field. The Hamiltonian for this model is ℋ==JΣ(ij)σiσj−ΣiHiσi, where σi,σj=±1, J is the nearest-neighbor interaction constant, and the field Hi=tH is fixed at each site with ti=±1 at random and Σti=0. L×L×L lattices with periodic boundary conditions have been studied for a range of H and T. As expected we find a ferromagnetic ordered state which for small H undergoes a second order phase change to the paramagnetic state with increasing temperature. A finite size scaling analysis of the preliminary data suggests that the critical exponent β is substantially smaller (β∼0.2) than the usual 3-dim Ising value of 0.31. Results obtained for small lattices indicate that below kT/J-2 the transition becomes first order suggesting that a tricritical point appears on the critcal field curve.