Site Percolation Threshold Research Articles

Critical behavior of the site diluted quantum anisotropic Heisenberg model in two dimensions

In this work we use the Self Consistent Harmonic Approximation and Quantum Monte Carlo technique to study the Quantum XY on a two dimensional square lattice in the presence of nonmagnetic impurities. In particular we discuss how site disorder changes the Berezinskii–Kosterlitz–Thouless transition temperature, TBKT. This temperature is determined as a function of the nonmagnetic density. Our results are consistent with an anomalous behavior of TBKT at a concentration close to the site percolation threshold. We interpret the results as due to a competition between the confining of vortices and quantum fluctuations, or due to finite size effects.

Multistage ordering and critical singularities inCo1−xZnxAl2O4(0≤x≤1): Dilution and pressure effects in a magnetically frustrated system

We report comprehensive studies of the crystallographic,magnetic, and thermal properties of a spinel-type magnetically frustrated compound, CoAl2O4, and a magnetically diluted system,Co1-xZnxAl2O4. These studies revealed the effects of dilution and disorder when the tetrahedral magnetic Co ion was replaced by the nonmagnetic Zn ion. Low-temperature anomalies were observed in magnetic susceptibility at x < 0.6. A multicritical point was apparent at T = 3.4 K and x = 0.12, where the antiferromagnetic, spin glass-like, and paramagnetic phases met. At that point, the quenched ferromagnetic component induced by a magnetic field during cooling was sharply enhanced and was observable below x = 0.6. Around x = 0.6, magnetic susceptibility and specific heat were described by temperature power laws, chi ~ C/T ~ T-d, in accord with the site percolation threshold of the diamond lattice. This behavior is reminiscent of a quantum critical singularity. We propose an x-temperature phase diagram in the range 0 < x < 1 for Co1-xZnxAl2O4. The transition temperature of CoAl2O4 determined from magnetic susceptibility measured under hydrostatic pressure increased with increasing pressure.

Universal aspects of critical percolation on random half-planar maps

We study a large class of Bernoulli percolation models on random lattices of the half-plane, obtained as local limits of uniform planar triangulations or quadrangulations. We first compute the exact value of the site percolation threshold in the quadrangular case using the so-called peeling techniques. Then, we generalize a result of Angel about the scaling limit of crossing probabilities, that are a natural analogue to Cardy’s formula in (non-random) plane lattices. Our main result is that those probabilities are universal, in the sense that they do not depend on the percolation model neither on the degree of the faces of the map.

Study of percolation and modeling of the order–disorder transition for zincblende–diamond structures: Percolation and the existence of a unique band of events

In this paper, the dependence of the crystallographic direction in the critical concentration of IV-type atoms for the zincblende–diamond (order–disorder) transition in (III–V)–IV semiconductor alloys were found for the (001), (110), and (111) growth directions by using Monte Carlo simulations. Intervals for the concentration values were determined by simulation, generating a band that is consistent with experimental data. Selecting the average value of this band produces good agreement with previous experimental results. It was also found that the relationship between order parameter and concentration does not have unique values and that there exists a band of events. The existence of this band can explain the scatter in the experimental data reported in x-ray, Raman, and optical anisotropy. It was found that the site percolation threshold is independent of the crystallographic direction of growth and converges to a single concentration value x ≈ 0.437.

Tight lower bound for percolation threshold on an infinite graph.

We construct a tight lower bound for the site percolation threshold on an infinite graph, which becomes exact for an infinite tree. The bound is given by the inverse of the maximal eigenvalue of the Hashimoto matrix used to count nonbacktracking walks on the original graph. Our bound always exceeds the inverse spectral radius of the graph's adjacency matrix, and it is also generally tighter than the existing bound in terms of the maximum degree. We give a constructive proof for existence of such an eigenvalue in the case of a connected infinite quasitransitive graph, a graph-theoretic analog of a translationally invariant system.

Magnetism in spin models for depleted honeycomb-lattice iridates: Spin-glass order towards percolation

Iridates are characterized by a fascinating interplay of spin-orbit and electron-electron interactions. The honeycomb-lattice materials ${A}_{2}{\mathrm{IrO}}_{3}$ $(A=\text{Na},\text{Li})$ have been proposed to realize pseudospin-1/2 Mott insulating states with strongly anisotropic exchange interactions, described by the Heisenberg-Kitaev model, but other scenarios involving longer-range exchange interactions or more delocalized electrons have been put forward as well. Here we study the influence of nonmagnetic doping, i.e., depleted moments, on the magnetic properties of experimentally relevant variants of the Heisenberg-Kitaev and Heisenberg ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}$ models. We generically find that the zigzag order of the clean system is replaced, upon doping, by a spin-glass state with short-ranged zigzag correlations. We determine the spin-glass temperature as a function of the doping level and show that this quantity allows one to assess the importance of longer-range exchange interactions when the doping is driven across the site-percolation threshold of the honeycomb lattice.

Elastically controlled manipulation of the magnetic order parameter and the metal-insulator transition in a manganite thin film

We measured the temperature dependence of the saturation magnetization (Ms) of a (La1-xPrx)1-yCayMnO3 (x ~ 0.60, y ~ 0.33) film as a function of applied bending stress. Stress producing a compressive strain of -0.01% along the magnetic easy axis increased the Curie temperature by ~6 K and the metal-insulator-transition by ~4 K. Regardless of whether or not stress is applied to the film, magnetic ordering occurs at temperatures significantly higher than the metal-insulator-transition temperature. The magnetization of the sample at the temperature of the metal-insulator-transition is approximately the site percolation threshold for a two-dimensional spin lattice.

Embedding binary sequences into Bernoulli site percolation on [formula omitted

We investigate the problem of embedding infinite binary sequences into Bernoulli site percolation on Zd with parameter p. In 1995, I. Benjamini and H. Kesten proved that, for d⩾10 and p=1/2, all sequences can be embedded, almost surely. They conjectured that the same should hold for d⩾3. We consider d⩾3 and p∈(pc(d),1−pc(d)), where pc(d)<1/2 is the critical threshold for site percolation on Zd. We show that there exists an integer M=M(p), such that, a.s., every binary sequence, for which every run of consecutive 0s or 1s contains at least M digits, can be embedded.

Spin transport in the two-dimensional quantum disordered anisotropic Heisenberg model

We use the self consistent harmonic approximation together with the Linear Response Theory to study the effect of nonmagnetic disorder on spin transport in the quantum diluted two-dimensional anisotropic Heisenberg model with spin S=1 in a square lattice. The model has a BKT transition at zero dilution. We calculate the regular part of the spin conductivity σreg(ω) and the Drude weight DS(T) as a function of the non-magnetic concentration, x. Our calculations show that the spin conductivity drops abruptly to zero at xcSCHA≈0.5 indicating that the system changes from an ideal spin conductor state to an insulator. This value is far above the site percolation threshold xcsite≈0.41. Although the SCHA fails in determining precisely the percolation threshold, both the spin conductivity and the Drude weight show a quite regular behavior inside 0≤x≤xcSCHA indicating that the transition stays in the same universality class all along the interval.

Site percolation on lattices with low average coordination numbers

.We present a study of site and bond percolation on periodic lattices with (on average) fewer than three nearest neighbors per site. We have studied this issue in two contexts: by simulating oxides with a mixture of 2-coordinated and higher-coordinated sites and by mapping site-bond percolation results onto a site model with mixed coordination number. Our results show that a conjectured power-law relationship between coordination number and site percolation threshold holds approximately if the coordination number is defined as the average number of connections available between high-coordinated sites and suggest that the conjectured power-law relationship reflects a real phenomenon requiring further study. The solution may be to modify the power-law relationship to be an implicit formula for percolation threshold, one that takes into account aspects of the lattice beyond spatial dimension and average coordination number.

Electroactive mixed self-assembled monolayers: A numerical overview of phase segregations

We propose a modeling of phase segregations, inspired by image filtering, and dedicated to electrochemical systems. This original approach is compared to previous experimental results and enhanced with the concepts of 2D segregation and site percolation threshold.

Rigorous Confidence Intervals on Critical Thresholds in 3 Dimensions

We extend the method of Balister, Bollobás and Walters (Phys. Rev. E 76:011110, 2007) for determining rigorous confidence intervals for the critical threshold of two dimensional lattices to three (and higher) dimensional lattices. We describe a method for determining a full confidence interval and apply it to show that the critical threshold for bond percolation on the simple cubic lattice is between \(0.2485\) and \(0.2490\) with \(99.9999\,\%\) confidence, and the critical threshold for site percolation on the same lattice is between \(0.3110\) and \(0.3118\) with \(99.9999\,\%\) confidence.

Site-percolation threshold of carbon nanotube fibers—Fast inspection of percolation with Markov stochastic theory

We present a site-percolation model based on a modified FCC lattice, as well as an efficient algorithm of inspecting percolation which takes advantage of the Markov stochastic theory, in order to study the percolation threshold of carbon nanotube (CNT) fibers. Our Markov-chain based algorithm carries out the inspection of percolation by performing repeated sparse matrix–vector multiplications, which allows parallelized computation to accelerate the inspection for a given configuration. With this approach, we determine that the site-percolation transition of CNT fibers occurs at pc=0.1533±0.0013, and analyze the dependence of the effective percolation threshold (corresponding to 0.5 percolation probability) on the length and the aspect ratio of a CNT fiber on a finite-size-scaling basis. We also discuss the aspect ratio dependence of percolation probability with various values of p (not restricted to pc).

High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials

The critical curves of the q-state Potts model can be determined exactly for regular two-dimensional lattices G that are of the three-terminal type. This comprises the square, triangular, hexagonal and bow–tie lattices. Jacobsen and Scullard have defined a graph polynomial PB(q, v) that gives access to the critical manifold for general lattices. It depends on a finite repeating part of the lattice, called the basis B, and its real roots in the temperature variable v = eK − 1 provide increasingly accurate approximations to the critical manifolds upon increasing the size of B. Using transfer matrix techniques, these authors computed PB(q, v) for large bases (up to 243 edges), obtaining determinations of the ferromagnetic critical point vc > 0 for the (4, 82), kagome, and (3, 122) lattices to a precision (of the order 10−8) slightly superior to that of the best available Monte Carlo simulations. In this paper we describe a more efficient transfer matrix approach to the computation of PB(q, v) that relies on a formulation within the periodic Temperley–Lieb algebra. This makes possible computations for substantially larger bases (up to 882 edges), and the precision on vc is hence taken to the range 10−13. We further show that a large variety of regular lattices can be cast in a form suitable for this approach. This includes all Archimedean lattices, their duals and their medials. For all these lattices we tabulate high-precision estimates of the bond percolation thresholds pc and Potts critical points vc. We also trace and discuss the full Potts critical manifold in the (q, v) plane, paying special attention to the antiferromagnetic region v < 0. Finally, we adapt the technique to site percolation as well, and compute the polynomials PB(p) for certain Archimedean and dual lattices (those having only cubic and quartic vertices), using very large bases (up to 243 vertices). This produces the site percolation thresholds pc to a precision of the order of 10−9.

Passivation in Non-Random Solid-Solution Alloys

Results from large-cell Monte Carlo renormalization group calculations for the effects of ordering and clustering on site percolation thresholds in bcc alloys are presented. The results demonstrate that for the case of relatively small positive heats of mixing (0.015 eV/atom), the site percolation threshold is lowered from its conventional value of ~0.25 to 0.16. Conversely, for a small negative heat of mixing (-0.015 eV/atom), the threshold increases to ~0.30. The implications of these results on corrosion/passivation in the context of the integrity of materials for nuclear power systems are indicated.

Inhomogeneous magnetism in the doped kagome lattice of LaCuO2.66

The hole-doped kagome lattice of Cu${}^{2+}$ ions in LaCuO${}_{2.66}$ was investigated by nuclear quadrupole resonance, electron spin resonance, electrical resistivity, bulk magnetization, and specific-heat measurements. For temperatures above $\ensuremath{\sim}180$ K, the spin and charge properties show an activated behavior suggestive of a narrow-gap semiconductor. At lower temperatures, the results indicate an insulating ground state which may or may not be charge ordered. While the frustrated spins in remaining patches of the original kagome lattice might not be directly detected here, the observation of coexisting nonmagnetic sites, free spins, and frozen moments reveals an intrinsically inhomogeneous magnetism. Numerical simulations of a $\frac{1}{3}$-diluted kagome lattice rationalize this magnetic state in terms of a heterogeneous distribution of cluster sizes and morphologies near the site-percolation threshold.

Percolation thresholds on three-dimensional lattices with three nearest neighbors

We present a study of site and bond percolation on periodic lattices with three nearest neighbors per site. Essentially all previous studies of percolation in 3D have considered coordination numbers of 4 or higher, but 3-coordinated lattices have attracted recent interest for their unusual symmetries and relevance to self-assembled materials. We have considered four lattices, with different symmetries, different underlying Bravais lattices, and different degrees of longer-range connections. As expected, we find that the site and bond percolation thresholds in all of the 3-connected lattices studied here are significantly higher than in the diamond lattice (a 4-connected lattice). Thresholds for different lattices are similar to within a few per cent, despite the differences between the lattices at scales beyond nearest and next-nearest neighbors. They also confirm an approximate analytical result for the relationship between coordination number and percolation threshold, one that had previously only been compared with simulation results for coordination numbers of 4 or higher.

Percolation thresholds on planar Euclidean relative-neighborhood graphs

In the present article, statistical properties regarding the topology and standard percolation on relative neighborhood graphs (RNGs) for planar sets of points, considering the Euclidean metric, are put under scrutiny. RNGs belong to the family of "proximity graphs"; i.e., their edgeset encodes proximity information regarding the close neighbors for the terminal nodes of a given edge. Therefore they are, e.g., discussed in the context of the construction of backbones for wireless ad hoc networks that guarantee connectedness of all underlying nodes. Here, by means of numerical simulations, we determine the asymptotic degree and diameter of RNGs and we estimate their bond and site percolation thresholds, which were previously conjectured to be nontrivial. We compare the results to regular 2D graphs for which the degree is close to that of the RNG. Finally, we deduce the common percolation critical exponents from the RNG data to verify that the associated universality class is that of standard 2D percolation.

Crossover behaviors in one and two dimensional heterogeneous load sharing fiber bundle models

We study the effect of heterogeneous load sharing in the fiber bundle models of fracture. The system is divided into two groups of fibers (fraction p and 1 − p) in which one group follows the completely local load sharing mechanism and the other group follows global load sharing mechanism. Patches of local disorders (weakness) in the loading plate can cause such a situation in the system. We find that in 2d a finite crossover (between global and local load sharing behaviours) point comes up at a finite value of the disorder concentration (near p c ∼ 0.53), which is slightly below the site percolation threshold. We numerically determine the phase diagrams (in 1d and 2d) and identify the critical behavior below p c with the mean field behavior (completely global load sharing) for both dimensions. This crossover can occur due to geometrical percolation of disorders in the loading plate. We also show how the critical point depends on the loading history, which is identified as a special property of local load sharing.

Berezinskii–Kosterlitz–Thouless transition close to the percolation threshold

Abstract We use Monte Carlo to investigate the Berezinskii–Kosterlitz–Thouless transition close to the site percolation threshold in a square lattice. Several thermodynamic quantities are calculated for lattice sizes L × L , from 16 L 640 . Our results are consistent with an infinite order transition for any value of the concentration of magnetic sites. We found that close to the critical percolation concentration, p c (0.592746), the Berezinskii–Kosterlitz–Thouless transition temperature goes to zero as T BKT ∝ ( p − p c ) 0.908 and the specific heat behaves as T s h ∝ p 1.133 .