Correlated Percolation Problem Research Articles

Ferromagnetic percolation transition in a multiorbital flat band assisted by Hund's coupling

By connecting Hund's physics with flat band physics, we establish an exact result for studying ferromagnetism in a multiorbital system. We consider a two-layer model consisting of a ${p}_{x}, {p}_{y}$-orbital honeycomb lattice layer and an $f$-orbital triangular lattice layer with sites aligned with the centers of the honeycomb plaquettes. The system features a flat band that admits a percolation representation for an appropriate chemical potential difference between the two layers. In this representation, the ground-state space is spanned by maximum-spin clusters of localized single-particle states, and averaging over the ground states yields a correlated percolation problem with weights due to the spin degeneracy of the clusters. A paramagnetic-ferromagnetic transition occurs as the band approaches half filling and the ground states become dominated by states with a large maximum-spin cluster, as shown by Monte Carlo simulation.

Disorder-Free Localization in an Interacting 2D Lattice Gauge Theory.

Disorder-free localization has been recently introduced as a mechanism for ergodicity breaking in low-dimensional homogeneous lattice gauge theories caused by local constraints imposed by gauge invariance. We show that also genuinely interacting systems in two spatial dimensions can become nonergodic as a consequence of this mechanism. This result is all the more surprising since the conventional many-body localization is conjectured to be unstable in two dimensions; hence the gauge invariance represents an alternative robust localization mechanism surviving in higher dimensions in the presence of interactions. Specifically, we demonstrate nonergodic behavior in the quantum link model by obtaining a bound on the localization-delocalization transition through a classical correlated percolation problem implying a fragmentation of Hilbert space on the nonergodic side of the transition. We study the quantum dynamics in this system by introducing the method of "variational classical networks," an efficient and perturbatively controlled representation of the wave function in terms of a network of classical spins akin to artificial neural networks. We identify a distinguishing dynamical signature by studying the propagation of line defects, yielding different light cone structures in the localized and ergodic phases, respectively. The methods we introduce in this work can be applied to any lattice gauge theory with finite-dimensional local Hilbert spaces irrespective of spatial dimensionality.

Roughness of Brittle Fractures as a Correlated Percolation Problem

The morphology of brittle fracture surfaces are self affine with roughness exponents that may be classified into a small number of universality classes. We discuss these in light of the recent proposal that the self affinity is a manifestation of the fracture process being a correlated percolation process. We also study numerically with high precision the roughness exponent in the two-dimensional fuse model with disorder both in breaking thresholds and conductances of the fuses. Our results are consistent with the predictions of the correlated percolation theory.

Connectivity and percolation in simulated grain-boundary networks

Random percolation theory is a common basis for modelling intergranular phenomena such as cracking, corrosion or diffusion. However, crystallographic constraints in real microstructures dictate that grain boundaries are not assembled at random. In this work a Monte Carlo method is used to construct physically realistic networks composed of high-angle grain boundaries that are susceptible to intergranular attack, as well as twin-variant boundaries that are damage resistant. When crystallographic constraints are enforced, the simulated networks exhibit triple-junction distributions that agree with experiment and reveal the non-random nature of grain-boundary connectivity. The percolation threshold has been determined for several constrained boundary networks and is substantially different from the classical result of percolation theory; compared with a randomly assembled network, about 50-75% more resistant boundaries are required to break up the network of susceptible boundaries. Triple-junction distributions are also shown to capture many details of the correlated percolation problem and to provide a simple means of ranking microstructures.

Cluster Structure of Collapsing Polymers

In order to better understand the geometry of the polymer collapse transition, we study the distribution of geometric clusters made up of the nearest neighbor interactions of an interacting self-avoiding walk. We argue for this new correlated percolation problem that in two dimensions, and possibly also in three dimensions, a percolation transition takes place at a temperature lower than the collapse transition. Hence this novel transition should be governed by exponents unrelated to the θ-point exponents. This also implies that there is a temperature range in which the polymer has collapsed, but has no long-range cluster structure. We use Monte Carlo to study the distribution of clusters on the simple cubic and Manhattan lattices. On the Manhattan lattice, where the data are most convincing, we find that the percolation transition occurs at ωp=1.461(3), while the collapse transition is known to occur exactly at ωθ=1.414.... We propose a finite-size scaling form for the cluster distribution and estimate several of the critical exponents. Regardless of the value of ωp, this percolation problem sheds new light on polymer collapse.

Growth of Cayley and diluted Cayley trees with two kinds of entities

A kinetic growth model derived from the magnetic Eden model is introduced in order to simulate the growth of hierarchical structures, such as Cayley trees. We only consider the case where two kinds of entities are competing with each other and can be further subjected to an external field. The very relevant case in which both kinds of entities have different coordination numbers is introduced here for the first time, and is called the diluted Cayley tree. Physical and geometrical properties of the finite and infinite trees are exactly found and simulated. Finite-size effects are emphasized and illustrated on the global or local magnetization and on the chemical activity. Asymptotic limits are given in each case. The generated patterns can be related to a correlated percolation problem briefly discussed in the appendix.

Annealed percolation: Determination of exponents in a correlated-percolation problem.

We describe a two-dimensional correlated-percolation model, which we call annealed percolation. In annealed percolation, attractive interactions between near-neighbor occupied sites produce a structure reminiscent of discontinuous metal films. We have carefully determined the percolation threshold p c and four exponents for this model: the correlation-length exponent ν, the cluster-size-distribution exponent τ, the fractal dimension of the infinite cluster d f , and the electrical-conductivity exponent t. We find that all these exponents have the same values as in pure (uncorrelated) percolation. We also compute the electrical conductance G versus concentration p for this model and compare with the G vs p curves for pure-site and pure-bond percolation models

Percolation, statistical topography, and transport in random media

A review of classical percolation theory is presented, with an emphasis on novel applications to statistical topography, turbulent diffusion, and heterogeneous media. Statistical topography involves the geometrical properties of the isosets (contour lines or surfaces) of a random potential $\ensuremath{\psi}(\mathrm{x})$. For rapidly decaying correlations of $\ensuremath{\psi}$, the isopotentials fall into the same universality class as the perimeters of percolation clusters. The topography of long-range correlated potentials involves many length scales and is associated either with the correlated percolation problem or with Mandelbrot's fractional Brownian reliefs. In all cases, the concept of fractal dimension is particularly fruitful in characterizing the geometry of random fields. The physical applications of statistical topography include diffusion in random velocity fields, heat and particle transport in turbulent plasmas, quantum Hall effect, magnetoresistance in inhomogeneous conductors with the classical Hall effect, and many others where random isopotentials are relevant. A geometrical approach to studying transport in random media, which captures essential qualitative features of the described phenomena, is advocated.

Thermodynamic consideration of the sol-gel transition in polymer solutions

The thermoreversible solgel transition in polymer solutions is discussed in terms of (i) liquid-liquid phase separation by spinodal decomposition, (ii) crystallite formation of polymer chains, and (iii) a hydrogenbonding type association of the polymer chains. The first two effects can be described by the familiar Flory-Huggins theory and the last by a modern theory of gelation as a result of the site-bond correlated percolation problem. The three equilibrium aspects interpret well our recent results on the solgel transition in poly(vinyl alcohol)/water and poly(vinyl chloride)/γ-butyrolactone systems.

Site-bond correlated percolation in a ferromagnetic and antiferromagnetic lattice gas: the Bethe cluster approximation

The authors study the site-bond correlated percolation problem in a lattice gas within the Bethe cluster approximation and obtain explicit results both in the ferromagnetic and antiferromagnetic regions.

Site bond percolation in ferromagnetic and antiferromagnetic Ising models: a renormalisation group approach

The Migdal-Kadanoff (1976) infinitesimal renormalisation group is employed to study the site bond correlated percolation problem, in a ferromagnetic and antiferromagnetic lattice gas. The general phase diagram shows that percolation is favoured by ferromagnetic interaction, while inhibited by antiferromagnetic interaction. Moreover, the lattice gas critical point is never a percolation point in the antiferromagnetic case, while it is a percolation point in the ferromagnetic case for a particular range of bond probability pB.

Clusters and Ising critical droplets: a renormalisation group approach

The Migdal-Kadanoff renormalisation group for two-dimensions is employed to obtain the global phase diagram for the site-bond correlated percolation problem. It is found that the Ising critical point (K=Kc,H=O) is a percolation point for a range of bond probability rho B such that 1>or= rho B>or=1-e-2Kc. In particular, as rho B approaches 1-e-2Kc, the percolation clusters become less compact and coincide with the Ising critical droplets.

Monte Carlo tests of universality in a correlated-site percolation problem

The authors study a correlated-site percolation model on a square lattice in which a site is occupied if all four bonds surrounding it are occupied; this model is of possible relevance to supercooled H2O and D2O. Using Monte Carlo methods, they find the critical concentration of sites rho c=0.562+or-0.001, and the critical exponents nu =1.33+or-0.07 and gamma =2.56+or-0.27. These results indicate that this correlated percolation problem is in the same universality class as uncorrelated percolation, while the critical threshold is decreased by about 5% relative to its value for the corresponding random-site percolation problems.