Kinetic Ising models with various single-spin-flip dynamics on quenched and annealed random regular graphs

We investigate a kinetic Ising model with several single-spin-flip dynamics (including Metropolis and heat bath) on quenched and annealed random regular graphs. As expected, on the quenched structures all proposed algorithms reproduce the same results since the conditions for the detailed balance and the Boltzmann distribution in an equilibrium are satisfied. However, on the annealed graphs the situation is far less clear—the network annealing disturbs the equilibrium moving the system away from it. Consequently, distinct dynamics lead to different steady states. We show that some algorithms are more resistant to the annealed disorder, which causes only small quantitative changes in the model behavior. On the other hand, there are dynamics for which the influence of annealing on the system is significant, and qualitative changes arise like switching the type of phase transition from a continuous to a discontinuous one. We try to identify features of the proposed dynamics which are responsible for the above phenomenon.

Figure 1

Phase diagrams for the Ising model on annelad and quenched $q$-RRG with $N={10}^5$ nodes and different degrees: (a) $q=4$, (b) $q=5$, (c) $q=6$, (d) $q=7$. Lines indicate analytical predictions of the mean-field approximation in the case of annealing: lines that are solid (stable states) for all values of $T$ correspond to heat-bath dynamics (HB annealed) and lines that are partially dashed (unstable states) to Metropolis (M annealed). Markers represent outcomes of Monte Carlo simulations: $•$ M annealed, $▴$ HB annealed, $\circ$ M quenched, $\times$ HB quenched. The critical temperature $T_c$ predicted by Eq. (8) almost exactly concurs with the simulated one on quenched graphs. The error bars are unnoticeable in a given scale.

Figure 2

(a) Transition temperature $T^{*}$ as a function of $q$ for the kinetic Ising model on the annealed $q$-RRG with two dynamics: $•$ Metropolis and $\times$ heat bath. Markers $\circ$ represent the results obtained from Eq. (8) for the quenched structure. Note that in the case of quenching, both dynamics give the same critical temperature. Moreover, the HB algorithm reproduces roughly the same results for both types of graphs, whereas the M algorithm gives a different value of $T^{*}$ on the annealed and quenched networks. Note that for the discontinuous phase transition $T^{*}$ is determined by the point at which three potential minima are equal; see also Fig. 5. (b) Transition rates as functions of $\mathrm{\Delta }E$ for heat-bath and modified Metropolis dynamics. Note that when $W_0=1$, the modified Metropolis algorithm coincides with the original version.

Figure 3

Transition rates as functions of $\mathrm{\Delta }E$ for different dynamics: $•W_{\text{M}}, \times W_{tanh}$ with $a=3, \diamond W_{tanh}$ with $a=1$, and $▴W_{exp}$. Consecutive panels correspond to higher temperatures: (a) $T=1$, (b) $T=3$, (c) $T=5$, and (d) $T=10$.

Figure 4

(a) Phase diagrams for the Ising model with the modified Metropolis algorithm Eq. (20) on annealed $q$-RRG for $W_0=0$. Markers represent outcomes of Monte Carlo simulations for the network comprised of $N={10}^5$ nodes: $•q=4, ▵q=5, \times q=6$, and $\circ q=7$. Lines indicate analytical predictions of the mean-field approximation. Solid ones correspond to stable states, and dashed ones refer to unstable points. (b) Phase diagrams for the Ising model with ${\text{M}}^{\prime }$ dynamics and $q=4$ for different values of $W_0$. Lighter lines correspond to higher $W_0$.

Figure 5

Phase diagram for the nonequilibrium kinetic Ising model on the annealed $q$-RRG with the modified Metropolis dynamics for $q=4$. Along the solid line, a phase transition occurs: below the tricritical point (TCP) there is a continuous phase transition and above TCP the continuous transition line splits up into two spinodal (dashed) lines, where the transition becomes first order. Along the dotted line, the term with $m^4$ in the Landau potential vanishes. The diagram divides a phase space into three distinctive areas: F where ferromagnetic (ordered) phase is present, P with paramagnetic (disordered) phase, and C a region where both phases coexist. Additionally, a schematic Landau potential is sketched for each area.

Figure 6

The dependence between the type of the phase transition and model's parameters for (a) the modified Metropolis dynamics and (b) the dynamics with $W_{tanh}$ rates. There are two characteristic regimes: with continuous (marked by oblique lines) and discontinuous phase transitions. In the case of discontinuous transitions, darker areas indicate wider hysteresis $\mathrm{\Delta }T$, measured as the difference between the upper and the lower spinodal lines. Note that in (a) the vertical section along $W_0=1$ corresponds to the original Metropolis dynamics, studied in [1]. For even values of interacting neighbors $q$, there exists a critical value of $W_0$ where discontinuous phase transition becomes continuous. Similar behavior is observed for the model with $W_{tanh}$ transition probabilities for odd $q$ values (b), but this time below a certain level of $a$, all transitions convert into continuous ones.