Threshold Exponents Research Articles

Phase transitions in the node, edge, bootstrap, and diffusion percolation models on the Sierpiński carpet

We investigate four types of percolation models — node, edge, bootstrap, and diffusion percolation — in three fractal graphs constructed on the Sierpiński carpet, employing the Monte Carlo method based on the Newman–Ziff algorithm. For each case, we calculate the percolation threshold and critical exponents (ν, γ, and β) through the crossing of percolation probabilities and the finite-size scaling analysis, incorporating correction-to-scaling effects. Our results reveal that critical exponents of the percolation phase transition in the three fractal graphs exhibit universality across all four percolation models. Furthermore, we demonstrate that the hyperscaling relation dν=γ+2β is also valid in the percolation phase transition on the Sierpiński carpet if the spatial dimension d is replaced by the Hausdorff dimension.

Percolative proton transport in hexagonal boron nitride membranes with edge-functionalization.

Two-dimensional layered materials have been used as matrices to study the structure and dynamics of trapped water and ions. Here, we demonstrate unique features of proton transport in layered hexagonal boron nitride membranes with edge-functionalization subject to hydration. The hydration-independent interlayer spacing indicates the absence of water intercalation between the h-BN sheets. An 18-fold increase in water sorption is observed upon amine functionalization of h-BN sheet edges. A 7-orders of magnitude increase in proton conductivity is observed with less than 5% water loading attributable to edge-conduction channels. The extremely low percolation threshold and non-universal critical exponents (2.90 ≤ α ≤ 4.43), are clear signatures of transport along the functionalized edges. Anomalous thickness dependence of conductivity is observed and its plausible origin is discussed.

Random adsorption process of linear k-mers on square lattices under the Achlioptas process.

We study the explosive percolation with k-mer random sequential adsorption (RSA) process. We consider both the Achlioptas process (AP) and the inverse Achlioptas process (IAP), in which giant cluster formation is prohibited and accelerated, respectively. By employing finite-size scaling analysis, we confirm that the percolation transitions are continuous, and thus we calculate the percolation threshold and critical exponents. This allows us to determine the universality class of the k-mer explosive percolation transition. Interestingly, the numerical simulation suggests that the universality class of the explosive percolation transition with the AP alters when the k-mer size changes. In contrast, the universality class of the transition with the IAP is independent of k, but it differs from that of the RSA without the IAP.

Percolation in metal-insulator composites of randomly packed spherocylindrical nanoparticles

While classical percolation is well understood, percolation effects in randomly packed or jammed structures are much less explored. Here we investigate both experimentally and theoretically the electrical percolation in a binary composite system of disordered spherocylinders, to identify the relation between structural (percolation) and functional properties of nanocomposites. Experimentally, we determine the percolation threshold $p_c$ and the conductivity critical exponent $t$ for composites of conducting (CrO$_2$) and insulating (Cr$_2$O$_3$) rodlike nanoparticles that are nominally geometrically identical, yielding $p_c=0.305 \pm 0.026$ and $t=2.52 \pm 0.03$ respectively. Simulations and modeling are implemented through a combination of the mechanical contraction method and a variant of random walk (de Gennes ant) approach, in which charge diffusion is correlated with the system conductivity via the Nernst-Einstein relation. The percolation threshold and critical exponents identified through finite size scaling are in good agreement with the experimental values. Curiously, the calculated percolation threshold for spherocylinders with an aspect ratio of 6.5, $p_c=0.312 \pm 0.002$, is very close (within numerical errors) to the one found previously in two other distinct systems of disordered jammed spheres and simple cubic lattice, an intriguing and surprising result.

The Helmholtz–Weyl decomposition of $$L^r$$ vector fields for two dimensional exterior domains

Let $$\Omega $$ be a two-dimensional exterior domain with smooth boundary $$\partial \Omega $$ and $$1< r < \infty $$ . Then $$L^r(\Omega )^2$$ allows a Helmholtz–Weyl decomposition, i.e., for every $$\mathbf{u}\in L^r(\Omega )^2$$ there exist $$\mathbf{h} \in X^r_{\tiny {\text{ har }}}(\Omega )$$ , $$w \in {\dot{H}}^{1,r}(\Omega )$$ and $$p \in {\dot{H}}^{1,r}(\Omega )$$ such that $$\begin{aligned} \mathbf{u} = \mathbf{h} + \mathrm{rot}\, w + \nabla p. \end{aligned}$$ The function $$\mathbf{h}$$ can be chosen alternatively also from $$V^r_{\tiny {\text{ har }}}(\Omega )$$ , another space of harmonic vector fields subject to different boundary conditions. These spaces $$X^r_{\tiny {\text{ har }}}(\Omega )$$ and $$V^r_{\tiny {\text{ har }}}(\Omega )$$ of harmonic vector fields are known to be finite dimensional. The above decomposition is unique if and only if $$1< r \leqq 2$$ , while in the case $$2< r < \infty $$ , uniqueness holds only modulo a one dimensional subspace of $$L^r(\Omega )^2$$ . The corresponding result for the three dimensional setting was proved in our previous paper, where in contrast to the two dimensional case, there are two threshold exponents, namely $$r=3/2$$ and $$r =3$$ . In our two dimensional situation, $$r=2$$ is the only critical exponent, which determines the validity of a unique Helmholtz–Weyl decomposition.

Bootstrap and diffusion percolation transitions in three-dimensional lattices

We study the bootstrap and diffusion percolation models in the simple-cubic (sc), body-centered cubic (bcc), and face-centered cubic (fcc) lattices using the Newman–Ziff algorithm. The percolation threshold and critical exponents were calculated through finite-size scaling with high precision in the three lattices. In addition to the continuous and first-order percolation transitions, we found a double transition, which is a continuous transition followed by a discontinuity of the order parameter. We show that the continuous transitions of the bootstrap and diffusion percolation models have the same critical exponents as the classical percolation within error bars and they all belong to the same universality class.

Newman-Ziff algorithm for the bootstrap percolation: Application to the Archimedean lattices

We propose very efficient algorithms for the bootstrap percolation and the diffusion percolation models by extending the Newman-Ziff algorithm of the classical percolation (M.E.J. Newman and R.M. Ziff (2000) [27]). Using these algorithms and the finite-size-scaling, we calculated with high precision the percolation threshold and critical exponents in the eleven two-dimensional Archimedean lattices. We present the condition for the continuous percolation transition in the bootstrap percolation and the diffusion percolation, and show that they have the same critical exponents as the classical percolation within error bars in two dimensions. We conclude that the bootstrap percolation and the diffusion percolation almost certainly belong to the same universality class as the classical percolation.

Major role of process conditions in tuning the percolation behavior of polyvinylidene fluoride based polymer/metal composites

The percolation behaviour of a polymer/metal composite (PMC), comprising polyvinylidene fluoride (PVDF)/nanocrystalline nickel (n-Ni) prepared by cold press and hot press methods, has been compared. Higher effective dielectric constants (εeff) with lower loss tangent (Tan δ) were observed for the cold pressed samples as compared to the hot pressed samples, which is attributed to better homogeneity and uniform distribution of n-Ni in the PVDF matrix. The percolation parameters [percolation threshold (fc) and critical exponents (s and s′)] are largely tuned due to the difference in process conditions. The non-universal fc, s, and s′ have been explained with the help of the percolation theory. PMC prepared though cold pressing would be a better candidate for static and low frequency dielectric applications.

Qualitative breakdown of the noncrossing approximation for the symmetric one-channel Anderson impurity model at all temperatures

The Anderson impurity model is studied by means of the self-consistent hybridization expansions in its non-crossing (NCA) and one-crossing (OCA) approximations. We have found that for the one-channel spin-$1/2$ particle-hole symmetric Anderson model, the NCA results are qualitatively wrong for any temperature, even when the approximation gives the exact threshold exponents of the ionic states. Actually, the NCA solution describes an overscreened Kondo effect, because it is the same as for the two-channel infinite-$U$ single level Anderson model. We explicitly show that the NCA is unable to distinguish between these two very different physical systems, independently of temperature. Using the impurity entropy as an example, we show that the low temperature values of the NCA entropy for the symmetric case yield the limit $S_{imp}(T=0)\rightarrow \ln\sqrt{2},$ which corresponds to the zero temperature entropy of the overscreened Kondo model. Similar pathologies are predicted for any other thermodynamic property. On the other hand, we have found that the OCA approach lifts the artificial mapping between the models and restores correct properties of the ground-state, for instance, a vanishing entropy at low enough temperatures $S_{imp}(T=0)\rightarrow0$. Our results indicate that the very well known NCA should be used with caution close to the symmetric point of the Anderson model.

Broadband dielectric spectroscopy of inhomogeneous and composite weak conductors

ABSTRACTIn this paper, we discuss broadband dielectric spectroscopy from mHz up to the infrared range mainly for materials with inhomogeneous weak conductivity, including conductor-dielectric nanocomposites. Our discussion is based on the effective medium approximation (EMA) and experiments modeled by this approach are reviewed. We discuss core–shell composites modeled by coated-spheres (Hashin–Shtrikman model) and normal composites with a possible percolation of the conductor component resulting in sharp or smeared percolation threshold of the DC conductivity and diverging static permittivity in the former case. The sharp percolation threshold is modeled by the Bruggeman EMA or by general EMA with arbitrary percolation threshold and arbitrary critical exponents of the DC conductivity and static permittivity. For the case of smeared percolation threshold in the case of complex topologies, we use the Lichtenecker model allowing for partial percolation of both the components. Finally, numerous papers reporting negative permittivity in weakly conducting materials are criticized and concluded to be due to spurious effects.

A transfer-matrix study of directed lattice animals and directed percolation on a square lattice

We studied the large-scale properties of directed lattice animals and directed percolation on a square lattice. Using a transfer-matrix approach on strips of finite widths, we generated relatively long sequences of estimates for effective values of critical fugacity, percolation threshold and correlation length critical exponents. We applied two different extrapolation methods to obtain estimates for infinite systems. The precision of our final estimates is comparable to (or better than) the precision of the best currently available results.

Thermal conductivity of Cu-graphite composites

Cu-graphite composites were prepared in the range of 0–40 vol.% of graphite. Thermal conductivity was measured using comparative cut bar method at room temperature (RT). The observed composition dependence is nonlinear. Mean field approximation with characteristic exponent equal to 1 and percolation threshold of copper phase 0.926 ± 0.039 are the nonlinear fitting results for enlarged experimental data range up to 50 vol.% of graphite. Average thermal conductivity of graphite 80 W/mK was estimated together with used graphite anisotropy values – 10 and 274 W/mK. It was observed that laser flash method gives 5% lower values of thermal conductivity at 30 vol.% of graphite than comparative cut bar method at RT. It was showed that observed percolation threshold and characteristic exponents for thermal and electrical conductivity dependence on composition for the investigated composite system are significantly different on the identical microstructure. It is due to phonon conduction via graphite phase in the case of thermal conductivity.

Percolation in the dielectric function of Pb(Zr, Ti)O3 – Pb2Ru2O6.5 ferroelectric – metal composites

A set of composite PbZr0.53Ti0.47O3 − xPb2Ru2O6.5 (PZT 53/47 − xPRO) and pure PbZr0.53Ti0.47O3 (PZT 53/47), PbZr0.38Ti0.62O3 (PZT 38/62) and PbZr0.36Ti0.64O3 (PZT 36/64) thick films (thicknesses 20–30 µm, porosity 25–50%) prepared by screen printing on (0001) single-crystal sapphire substrates, were studied using Fourier-transform infrared (FTIR) reflectivity and time-domain terahertz transmission spectroscopy. The compositions studied with 10, 15, 20 and 25 vol% PRO were around the electrical percolation threshold in the bulk ceramic composites (~17 vol%), known from impedance spectroscopy by Bobnar et al (2008 Appl. Phys. Lett. 92 182911). To obtain the effective dielectric functions of the films, we accounted for their porosity by using the Lichtenecker model. Using the dielectric functions of bulk pure PZT 53/47 ceramics and PRO single crystals from the literature available, effective dielectric functions were compared with those obtained using the generalized effective medium model by McLachlan with percolation threshold and critical percolation exponents, which fit the low-frequency data by Bobnar et al. The effective dynamic response in all the samples was dominated by an overdamped THz conductivity peak (central mode), as in pure PZT. The percolation threshold was not particularly apparent in the polar-phonon frequency range as it characterized the localized response. Unlike the low-frequency permittivity, which diverged at the percolation threshold, the effective microwave–terahertz permittivity increased above it.

Bond dimer percolation on square lattices

Percolation due to the simultaneous occupation of two neighboring bond sites, namely a bond dimer, is considered here by means of the renormalization cell technique providing an analytic way to obtain results such as percolation threshold, jamming coverage and critical exponents. This is complementary to previous numerical studies and extends the validation of the renormalization cell technique. Four different bond dimer depositions are considered: nematic, straight, angular and tortuous; results for each of them are given and analyzed separately. Size of the cells is varied. These results are combined with means of finite size scaling to obtain tendencies towards the thermodynamic limit. It is observed that the percolation threshold is reached at lower concentrations than for monomeric bond percolation establishing a trend for correlated bond percolation similar to the one already established for site dimer percolation. Two different techniques are used to obtain the percolation threshold getting results that are in good agreement with numerical simulations; similarly acceptable results for jamming coverage are obtained. Values for critical exponents are also in good agreement with those reported by means of numerical techniques.

Percolation of polyatomic species on a simple cubic lattice

In the present paper, the site-percolation problem corresponding to linear k-mers (containing k identical units, each one occupying a lattice site) on a simple cubic lattice has been studied. The k-mers were irreversibly and isotropically deposited into the lattice. Then, the percolation threshold and critical exponents were obtained by numerical simulations and finite-size scaling theory. The results, obtained for k ranging from 1 to 100, revealed that (i) the percolation threshold exhibits a decreasing function when it is plotted as a function of the k-mer size; and (ii) the phase transition occurring in the system belongs to the standard 3D percolation universality class regardless of the value of k considered.

Ground-state counterpropagating solitons in photorefractive media with saturable nonlinearity

We investigate the existence and form of (2+1)-dimensional ground-state counterpropagating solitons in photorefractive media with saturable nonlinearity. General conditions for the existence of fundamental solitons in a local isotropic model that includes an intensity-dependent saturable nonlinearity are identified. We confirm our theoretical findings numerically and determine the ground-state profiles. We check their stability in propagation and identify the coupling constant threshold for their existence. Critical exponents of the power and beam width are determined as functions of the propagation constant at the threshold. We finally formulate a variational approach to the same problem, introduce an approximate fundamental Gaussian solution, and verify that this method leads to the same threshold and similar critical exponents as the theoretical and numerical methods.

An Improved Wavelet Denoising Algorithm for Wideband Radar Targets Detection

A novel denoising technique based on wavelet transform modulus maxima (WTMM) is proposed for processing wideband radar spread targets detection signal in a clutter environment. Combined with the improved adaptive Bayes–Shrink threshold and Lipschitz exponents, we propose the path pruned approach at each scale terms as full-scale to split the signal. The estimation of WTMM over each scale has been optimized, thus, the signal and the noise can be split effectively. Additionally, to improve the computational efficiency, a fast method based on a piecewise polynomial interpolation algorithm is applied for the split signal reconstruction. Statistical results are quite promising and perform better than the conventional denoising algorithms: compared with the classical WTMM algorithm, the improved WTMM full-scale denoising algorithm not only increases the signal-to-noise (SNR) ratio by over 10 % but also reduces the processing time by 88 % and reduces the root-mean-square-error (RMSE) by over 35 %. More generally, the proposed algorithm has better performance than that of several typical algorithms in its denoising quality and singularity detection.

Spin-charge separation in one-dimensional fermion systems beyond Luttinger liquid theory

We develop a nonperturbative zero-temperature theory for the dynamic response functions of interacting one-dimensional spin-1/2 fermions. In contrast to the conventional Luttinger liquid theory, we take into account the nonlinearity of the fermion dispersion exactly. We calculate the power-law singularities of the spectral function and the charge and spin density structure factors for arbitrary momenta and interaction strengths. The exponents characterizing the singularities are functions of momenta and differ significantly from the predictions of the linear Luttinger liquid theory. We generalize the notion of the spin-charge separation to the nonlinear spectrum. This generalization leads to phenomenological relations between threshold exponents and the threshold energy.

Fate of 1D Spin-Charge Separation Away from Fermi Points

We consider the dynamic response functions of interacting one dimensional spin-1/2 fermions at arbitrary momenta. We build a nonperturbative zero-temperature theory of the threshold singularities using mobile impurity Hamiltonians. The interaction induced low-energy spin-charge separation and power-law threshold singularities survive away from Fermi points. We express the threshold exponents in terms of the spinon spectrum.

Effective transport properties of random composites: Continuum calculations versus mapping to a network

The effective transport properties and percolation of continuum composites have commonly been studied using discrete models, i.e., by mapping the continuum to a lattice or network. In this study we instead directly solve the continuum transport equations for composite microstructures both analytically and numerically, and we extract the continuum percolation threshold and scaling exponents for the two-dimensional square tile system. We especially focus on the role of corner contacts on flux flow and further show that mapping such "random checkerboard" systems to a network leads to a spurious secondary percolation threshold and causes shifts in the critical scaling exponents of the effective transport properties.