Metastability in the diluted parallel Ising model

TL;DR

The work analyzes parallel kinetic Ising simulations, showing that parallel Glauber updates can yield a checkerboard-like decoupling of sublattices and potential ergodicity issues, whereas Wolff cluster updates avoid this problem by sampling the full factorized distribution in the symmetry-broken phase. Introducing a small asynchronous dilution parameter $p$ couples the sublattices and destabilizes the checkerboard state, but exhibits finite-size–dependent metastable transients with an exit time $\langle T_e\rangle$ that scales with system size and shows a possible critical dilution around $p_c\approx 0.028$. The study draws connections between dilution, direct coupling in the Hamiltonian, and external-field-like effects, framing metastability in parallel updates as akin to nucleation phenomena and with implications for disordered systems such as spin glasses where metastable states are prevalent. Overall, the results illuminate how parallelization choices and dilution can affect ergodicity and phase sampling, informing how to interpret metastable behavior in complex or disordered systems. The insights are relevant for designing parallel Monte Carlo schemes and for understanding metastability in systems where symmetry breaking is subtle or obscured.

Abstract

We present some considerations about the parallel implementations of the kinetic (Monte Carlo) version of the Ising model. In some cases the equilibrium distribution of the parallel version does not present the symmetry breaking phenomenon in the low-temperature phase, i.e., the stochastic trajectory originated by the Monte Carlo simulation can jump between the distributions corresponding to both kinds of magnetization, or the lattice can break into two disjoint sublattices, each of which goes into a different asymptotic distribution (phase). In this latter case, by introducing a small asynchronism (dilution), we can have a transition between the homogeneous and the checkerboard phases, with metastable transients.

Figures (8)

• Figure 1: (a) The separation of the two sublattices for the parallel Glauber dynamics and the in-place update possibility (after having copied the left-boundary spin). (b) Similar procedure for the two-dimensional lattice.
• Figure 2: (a) Time plot of the magnetization $M(t)$ for a $16\times 16$ serial Ising model for $J=0.45$ ($J_c\simeq 0.44$), time measured in Monte-Carlo steps. Due to the small lattice size, magnetization occasionally jumps from positive to negative values, since the asymptotic distribution is not fully factorized. (b) Time plot of the magnetization for a $256\times 256$ Ising model updated using the Wolff algorithm, $J=0.45$, time measured in cluster clips. Despite the large size of the lattice, in this case the magnetization often flips between positive and negative values. Notice that in the sub-figure (b) the time scale is much smaller than in the sub-figure (a).
• Figure 3: Typical patterns in a $50\times50$ lattice (axes $x$ and $y$), for $J=1$ and $t=10000$, where white spots marks negative spins and black spots positive ones. (a) The three phases in the fully parallel Ising model ($p=0$), starting from a disordered configuration. Asymptotically, only one phase (black, white or checkerboard) survives. (b) Stability (with fluctuations) of the checkerboard pattern (which is the initial state) for small dilutions ($p=0.02$). (c) Droplets growing for larger dilutions ($p=0.045$), again starting from a checkerboard pattern.
• Figure 4: Time behavior of correlation index $c$ vs time $t$ for $L=16$ and coupling (a) $J=0.40$ (below the critical value $J_c\simeq 0.44$, the configuration is disordered and the index $c$ fluctuates around zero), (b) $J=0.45$ (above the critical value $J_c\simeq 0.44$, the system oscillates from homogeneous ($c>0$) and checkerboard ($c<0$) configurations (phases). We use a small value of $L$ ($16$) so that these switch are easily detectable, in the large $N=L\times L$ limit only one phase survives.
• Figure 5: Probability distribution of the correlation index $c$ for $L=32$ and several values of $J$. The distribution is computed over one evolution for $2\times 10^6$ time steps after a transient of $10^4$ time steps and using 100 bins. For $J=0.50$ the probability distribution is not unique (we show two simulations with different asymptotic distributions). The jagged shape of the distribution is due to the binning procedure.