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Register a new userCriticality governed by the stable renormalization fixed point of the Ising model in the hierarchical small-world network
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We study the Ising model in a hierarchical small-world network by renormalization group analysis, and find a phase transition between an ordered phase and a critical phase, which is driven by the coupling strength of the shortcut edges. Unlike ordinary phase transitions, which are related to unstable renormalization fixed points (FPs), the singularity in the ordered phase of the present model is governed by the FP that coincides with the stable FP of the ordered phase. The weak stability of the FP yields peculiar criticalities including logarithmic behavior. On the other hand, the critical phase is related to a nontrivial FP, which depends on the coupling strength and is continuously connected to the ordered FP at the transition point. We show that this continuity indicates the existence of a finite correlation-length-like quantity inside the critical phase, which diverges upon approaching the transition point.
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Monte Carlo simulations are performed to study the two-dimensional Potts models with q=3 and 4 states on directed Small-World network. The disordered system is simulated applying the Heat bath Monte Carlo update algorithm. A first-order and second-order phase transition is found for q=3 depending on the rewiring probability $p$, but for q=4 the system presents only a first-order phase transition for any value $p$ . This critical behavior is different from the Potts model on a square lattice, where the second-order phase transition is present for $qle4$ and a first-order phase transition is present for q>4.
We study the collective behavior of an Ising system on a small-world network with the interaction $J(r) propto r^{-alpha}$, where $r$ represents the Euclidean distance between two nodes. In the case of $alpha = 0$ corresponding to the uniform interaction, the system is known to possess a phase transition of the mean-field nature, while the system with the short-range interaction $(alphatoinfty)$ does not exhibit long-range order at any finite temperature. Monte Carlo simulations are performed at various values of $alpha$, and the critical value $alpha_c$ beyond which the long-range order does not emerge is estimated to be zero. Thus concluded is the absence of a phase transition in the system with the algebraically decaying interaction $r^{-alpha}$ for any nonzero positive value of $alpha$.
We study bond percolation on a one-parameter family of hierarchical small-world network, and find a meta-transition between the inverted BKT transition and the abrupt transition driven by changing the network topology. It is found that the order parameter is continuous and fractal exponent is discontinuous in the inverted BKT transition, and oppositely, the former is discontinuous and the latter is continuous in the abrupt transition. The gaps of the order parameter and fractal exponent in each transition go to vanish as approaching the meta-transition point. This point corresponds to a marginal power-law transition. In the renormalization group formalism, this meta-transition corresponds to the transition between transcritical and saddle-node bifurcations of the fixed point via a pitchfork bifurcation.
We investigated the Ising model on a square lattice with ferro and antiferromagnetic interactions modulated by the quasiperiodic Octonacci sequence in both directions of the lattice. We have applied the Replica Exchange Monte Carlo (Parallel Tempering) technique to calculate the thermodynamic quantities of the system. We obtained the order parameter, the associated magnetic susceptibility ($chi$) and the specific heat $(c)$ in order to characterize the universality class of the phase transition. Also, we use the finite size scaling method to obtain the critical temperature of the system and the critical exponents $beta$, $gamma$ and $ u$. In the low temperature limit we have obtained a continuous transition with critical temperature around $T_{c} approx 1.413$. The system obeys the Ising universality class with logarithmic corrections. We found estimatives for the correction exponents $hat{beta}$, $hat{gamma}$ and $hat{lambda}$ by using the finite size scaling technique.
We consider the Ising model on a small-world network, where the long-range interaction strength $J_2$ is in general different from the local interaction strength $J_1$, and examine its relaxation behaviors as well as phase transitions. As $J_2/J_1$ is raised from zero, the critical temperature also increases, manifesting contributions of long-range interactions to ordering. However, it becomes saturated eventually at large values of $J_2/J_1$ and the system is found to display very slow relaxation, revealing that ordering dynamics is inhibited rather than facilitated by strong long-range interactions. To circumvent this problem, we propose a modified updating algorithm in Monte Carlo simulations, assisting the system to reach equilibrium quickly.
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