Cluster percolation in the three-dimensional $\pm J$ random-bond Ising model

TL;DR

This study investigates how geometric percolation of various spin-cluster definitions relates to thermal ordering in the three-dimensional ±J RBIM across 0 ≤ φ ≤ 0.5. By combining FKCK, Houdayer, and two-replica CM RJ cluster constructions with extensive parallel-tempering simulations and finite-size scaling, the authors reveal that FKCK and CM RJ percolation coincide with ordering transitions in the pure ferromagnet, while in frustrated regimes percolation thresholds shift above or between ordering temperatures and exhibit random-percolation universality. They show that the two largest CM RJ clusters typically percolate with equal density at high T, and the density differences between the top clusters correlate with the corresponding order parameters (magnetization or overlap), providing percolation-based signatures of ferromagnetic and spin-glass transitions. The work discusses conserved-overlap phenomena and the potential of multi-replica cluster definitions to further illuminate spin-glass physics, with implications for improved Monte Carlo updates and geometric perspectives on ordering phenomena.

Abstract

Based on extensive parallel-tempering Monte Carlo simulations, we investigate the relationship between cluster percolation and equilibrium ordering phenomena in the three-dimensional $\pm J$ random-bond Ising model as one varies the fraction of antiferromagnetic bonds. We consider a range of cluster definitions, most of which are constructed in the space of overlaps between two independent real replicas of the system. In the pure ferromagnet that is contained as a limiting case in the class of problems considered, the relevant percolation point coincides with the thermodynamic ordering transition. For the disordered ferromagnet encountered first on introducing antiferromagnetic bonds and the adjacent spin-glass phase of strong disorder this connection is altered, and one finds a percolation transition above the thermodynamic ordering point that is accompanied by the appearance of /two/ percolating clusters of equal density. Only at the lower (disordered) ferromagnetic or spin-glass transition points the densities of these two clusters start to diverge, thus providing a percolation signature of these thermodynamic transitions. We compare the scaling behavior at this secondary percolation transition with the thermodynamic behavior at the corresponding ferromagnetic and spin-glass phase transitions.

Figures (17)

• Figure 1: Phase diagram of the three-dimensional $\pm J$ random-bond Ising model according to the data of Refs. HasenbuschEtAl2007CriticalBehaviorOfThe3DPMJIsingModelAtTheParamagneticFerromagneticTransitionLineHasenbuschEtAl2008CriticalBehaviorOf3DIsingSGModelsHasenbuschEtAl2007MagneticGlassyMulticriticalBehaviorOfThe3DPMJIsingModelFajenHartmannYoung2020percolationPercolationOfFortuinKasteleynClustersForTheRandomBondIsingModelCeccarelliPelissettoVivari2011FerromagneticGlassyTransitionsIn3DIsingSGsCreightonKatzgraber2011SamplingTheGroundStateMagnetizationOfDDimensionalPBodyIsingModels. The data-points for the CMRJ percolation transition originate from the present study (green crosses). The red dot illustrates the location of the multicritical Nishimori point. The black dots indicate important transition points: the pure Ising critical point, the intersection of the FKCK transition line with the Nishimori line, the spin-glass transition temperature for $\phi=0.5$, and the zero-temperature transition between the ferromagnetic and spin-glass phases. The lines connecting the data points are included as visual guides, except for the Nishimori line, which is explicitly defined in Eq. \ref{['eq:nishimori_line']}.
• Figure 2: Densities of the two largest Ising clusters as a function of temperature for the pure ferromagnet using three exemplary system sizes, $L=8$, $L=32$ and $L=128$. Inset (a) shows the corresponding relationship for the Houdayer clusters. $T_{\mathrm{I}}$ denotes the critical temperature of the ferromagnetic phase transition. Inset (b) illustrates how the number of wrapping Ising clusters decreases from two to one at $T_{\mathrm{Icl}}$ (see main text for details). The system sizes plotted are $L = 32$, $64$, and $128$.
• Figure 3: Data collapse of the wrapping probability of CMRJ clusters, obtained according to Eq. \ref{['eq:fss_wrapping_probability']}, in the critical region of the CMRJ percolation transition for the pure Ising ferromagnet ($\phi = 0$), with $t=(T-T_\mathrm{CMRJ})/T_\mathrm{CMRJ}$. From this collapse, the critical parameters are estimated as $T_\mathrm{CMRJ}=4.511\,527(16)$ and $\nu=0.6300(17)$. The inset shows the unscaled data in the vicinity of the critical point.
• Figure 4: Data collapse of the connectivity length according to Eq. \ref{['eq:cluster_connectivity_length']} with $t=(T-T_\mathrm{CMRJ})/T_{\mathrm{CMRJ}}$ for the parameters $T_\mathrm{CMRJ}=4.511\,527$ and $\nu=0.6300$. The inset shows the corresponding data collapse for the correlation length of the magnetization, defined in Eq. \ref{['eq:second_moment_correlation_length']}, using the same parameters, i.e., $T_\mathrm{I}=4.511\,527$, $\nu=0.6300$, and $t=(T-T_\mathrm{I})/T_\mathrm{I}$.
• Figure 5: Data collapse of the density of the largest CMRJ cluster according to Eq. \ref{['eq:fss_largest_cluster_size']} for the pure Ising ferromagnet with $t=(T-T_\mathrm{CMRJ})/T_\mathrm{CMRJ}$. From this collapse, the values $T_{\mathrm{CMRJ}} = 4.511\,534(26)$ and $\beta/\nu = 0.516(4)$ are obtained. The parameter $1/\nu$, with $\nu = 0.6300$, is held constant during the optimization. The inset illustrates four different quantities, all of which can serve as order parameters of the ferromagnetic phase transition.