Stochastic field effects in a two-state system: symmetry breaking and symmetry restoring

Abstract

We study the Ising model under a time-varying, but spatially homogeneous, Gaussian random magnetic field. In the Monte Carlo simulations, we go beyond the standard analysis of the order parameter by measuring the magnetization probability distribution as a function of temperature and field strength, and by computing the time required for the system to escape from a completely ordered state of the magnetization. We identify three distinct phases: a soft-paramagnetic phase, a soft-ferromagnetic phase and a bona-fide ferromagnetic phase. These soft phases display broad magnetization distributions that tend to limiting forms that remain finite in both height and width in the thermodynamic limit. The transition between the soft-paramagnetic and soft-ferromagnetic phases is a noise-induced transition and, for small field amplitudes, occurs at the critical temperature of the field-free Ising model. The transition from the soft-ferromagnetic to the ferromagnetic phase occurs at lower temperatures and is discontinuous, yet it does not fall into the conventional first-order class. Instead, it is characterized by a diverging escape time from an ordered magnetization state.

Figures (11)

• Figure 1: Soft-paramagnetic and soft- ferromagnetic phases. Histograms of the normalized probability distribution of the magnetization, $P(m)$, for different field intensities ($D = 0, 0.1, 0.6$) and for two system sizes ($L = 50, 200$) at several temperatures. Note that graphs in the same row have the same vertical scale. The results come from agent-based numerical simulations in which, after a thermalization time of $10^6$ MCS, we have let the system evolve for $10^8$ MCS and have measured the instantaneous magnetization every 10 MCS. Thus, the distributions $P(m)$ are derived from a sample of $10^7$ measurements of $m(t)$. These and subsequent histograms presented in this work have been computed using 500 bins of width $\Delta m=0.004$ in the interval $m\in[-1,1]$.
• Figure 2: Limiting wide distribution in the soft-ferromagnetic phase. Probability distribution of magnetization, $P(m)$, at temperature $T = 2.1$ for several system sizes and for two noise intensities ($D = 0.1, 0.6$). In the numerical simulations, we have taken $10^7$ measurements of the instantaneous magnetization after an initial transient of $10^6$ MCS. Note that, in the left panel, the distributions for $L=300$ and $L=500$ overlap at the scale of the figure.
• Figure 3: Ferromagnetic phase. Probability distribution of the magnetization, $P(m)$, for two system sizes ($L = 50, 200$) and for two field intensities ($D = 0.1, 0.6$) at different temperatures. Note that graphs in the same row have the same vertical scale. In the numerical simulations, the thermalization and measurement times have been set identical to those in Fig. \ref{['fig:HistogramsComparison']}.
• Figure 4: Region of validity of the different phases. Maxima $m_\mathrm{max}$ of the probability distribution of magnetization, $P(m)$, as a function of temperature for several intensities of the random magnetic field ($D = 0$, 0.1, 0.6) and for two system sizes ($L = 50, 200$). The black dashed lines correspond to the Onsager's theoretical solution, while symbols represent the outcomes from numerical simulations. Open circles represent the ferromagnetic phase, while colored circles correspond either to the soft-ferromagnetic phase (two circles located at two symmetric values of $m$) or to the paramagnetic phase---true or soft---(a single circle around $m = 0$). The same thermalization and measurement times as in Fig. \ref{['fig:HistogramsComparison']} have been applied here.
• Figure 5: Phase diagram in the $(D,T)$ plane. The diagram indicates the expected shape of the probability distribution of the magnetization, together with the corresponding order parameter, in the thermodynamic limit for each region. The arrows in the probability distribution indicate a Dirac-delta contribution. The horizontal dashed line corresponds to the critical temperature of the field-free Ising model, $T_{\mathrm{c}} \approx 2.269$. The colored lines separating the soft-ferromagnetic and ferromagnetic phases come from numerical simulations for a system of size $L = 100$. In the simulations, given a value of $D$, we look for the first temperature at which the system is able to jump from $m = -1$ to $m = 0$ in a time shorter than a given observation time $\tau_\text{max}$. We then identify this temperature with the temperature of separation between the soft-ferromagnetic and ferromagnetic phases for that combination of $D$ and $\tau_\text{max}$ values. The curves in the figure represent the average of 10 repetitions of this procedure.