Critical behaviour at the displacive limit of structural phase transitions

For a special critical point at zero temperature,T _c =0, which is called the displacive limit of a classical or of a quantum-mechanical model showing displacive phase transitions, we derive a set of static critical exponents in the large-n limit. Due to zero-point motions and quantum fluctuations at low temperatures, the exponents of the quantum model are different from those of the classical model. Moreover, we report results on scaling functions, corrections to scaling, and logarithmic factors which appear ford=2 in the classical case and ford=3 in the quantum-mechanical case. Zero-point motions cause a decrease of the critical temperature of the quantum model with respect to the classicalT _c , which implies a difference between the classical and the quantum displacive limit. However, finite critical temperatures are found in both cases ford>2, while critical fluctuations still occur atT _c =0 for 0<d≦2 in the classical case and for 1 <d≦2 in the quantum model. Further we derive the slope of the critical curve at the classical displacive limit exactly. The absence of 1/n-corrections to the exponents of the classical model is also discussed.     

Correlations in inhomogeneous Ising models

Using general methods developed in a previous treatment we study correlations in inhomogeneous Ising models on a square lattice. The nearest neighbour couplings can vary both in strength and sign such that the coupling distribution is translationally invariant in diagonal direction. We calculate correlations parallel to the layering in the diagonally layered model with periodv=2, the so-called “general square lattice” model (GS). If the model has a finite critical temperature,T _c>0, we have a spontaneous magnetization belowT _c vanishing atT _c with the Ising exponent β=1/8. AtT _c correlations decay algebraically with critical exponnet η=1/4 and exponentially forT>T _c. In the frustrated case we have oscillatory behaviour superposed on the exponential decay where the wavevector of the oscillations changes at some “disorder temperature”T _D(>T _c) from commensurate to temperature-dependent in commensurate periods. If the critical temperature vanishes,T _c=0 we always have exponential decay at finite temperatures, while atT=T _c=0 we encounter either long-range order or algebraic decay with critical index η=1/2, i.e.T=0 is thus a critical point.     

A-B site-bond correlated percolation for antiferromagnetic Ising model: The Bethe cluster approximation

We study the site-bond percolation problem for clusters of holes and particles with antiferromagnetic order by means of the Bethe cluster approximation. We find that the droplets (i.e.P _B =1?e ^?|K|/2) diverge at the antiferromagnetic critical pointH=0,T=T _c; however forH≠0 they diverge along a percolation line which is different from the Antiferromagnetic Phase Boundary except atT=0.     

Phase diagram for three-dimensional correlated site-bond percolation

The Coniglio-Stanley-Klein model is a random bond percolation process between the occupied sites of a lattice gas in thermal equilibrium. Our Monte Carlo simulation for 40³ and 60³ simple cubic lattices determines at which bond thresholdp _Bc , as a function of temperatureT and concentrationx of occupied sites, an infinite network of active bonds connects occupied sites. The curvesp _Bc (x, T) depend only slightly onT whereas they cross over if plotted as a function of the field conjugate tox. Except close toT _c we find 1/p _Bc to be approximated well by a linear function ofx, in the whole interval between the thresholdx _c (T) of interacting site percolation atp _Bc =1 and the random bond percolation limitx=1 atp _Bc =0.248±0.001. Thisx _c (T) varied between 0.22 forT=0.96 (coexistence curve) and 0.3117±0.0003 forT= (random site percolation). At the critical point (T=T andx=1/2) we confirmed quite accurately the predictionp _Bc =1-exp(–2J/k _B T _c ) of Coniglio and Klein. As a byproduct we found 0.89±0.01 for the critical exponent of the correlation length in random percolation.     

Directed spiral site percolation on the square lattice

A new site percolation model, directed spiral percolation (DSP), under both directional and rotational (spiral) constraints is studied numerically on the square lattice. The critical percolation threshold p _c ≈ 0.655 is found between the directed and spiral percolation thresholds. Infinite percolation clusters are fractals of dimension d _f ≈ 1.733. The clusters generated are anisotropic. Due to the rotational constraint, the cluster growth is deviated from that expected due to the directional constraint. Connectivity lengths, one along the elongation of the cluster and the other perpendicular to it, diverge as p → p _c with different critical exponents. The clusters are less anisotropic than the directed percolation clusters. Different moments of the cluster size distribution P _s(p) show power law behaviour with | p - p _c| in the critical regime with appropriate critical exponents. The values of the critical exponents are estimated and found to be very different from those obtained in other percolation models. The proposed DSP model thus belongs to a new universality class. A scaling theory has been developed for the cluster related quantities. The critical exponents satisfy the scaling relations including the hyperscaling which is violated in directed percolation. A reasonable data collapse is observed in favour of the assumed scaling function form of P _s(p). The results obtained are in good agreement with other model calculations. Received 10 November 2002 / Received in final form 20 February 2003 Published online 23 May 2003 RID="a" ID="a"e-mail: santra@iitg.ernet.in     

Critical behavior of short range Potts glasses

We study by means of Monte Carlo simulations and the numerical transfer matrix technique the critical behavior of the short rangep=3 state Potts glass model in dimensionsd=2,3,4 with both Gaussian and bimodal (±J) nearest neighbor interactions on hypercubic lattices employing finite size scaling ideas. Ind=2 in addition the degeneracy of the glass ground state is computed as a function of the number of Potts states forp=3, 4, 5 and compared to that of the antiferromagnetic ground state. Our data indicate a transition into a glass phase atT=0 ind=2 with an algebraic singularity, aT=0 transition ind=3 with an essential singularity of the form exp(const.T ^–2), and an algebraic singularity atT0.25 ind=4. We conclude that the lower critical dimension of the present model isd _c =3 or very close to it. Some of the critical exponents are estimated and their respective values discussed.     

Correlations in inhomogeneous Ising models

Using general methods developed in a previous treatment we study correlations in inhomogeneous Ising models on a square lattice. The nearest neighbour couplings can vary both in strength and sign such that the coupling distribution is translationally invariant in horizontal direction. We calculate correlations parallel to the layering in the horizontally layered model with periodv=2. If the model has a finite critical temperature,T _c>0, the order parameter in the frustrated case may become discontinuous forT0. Correlations atT=T _c decay algebraically with critical exponent =1/4 and exponentially forT>T _c. If the critical temperature vanishes,T _c=0, we always have exponential decay at finite temperatures, while atT=T _c=0 we encounter either long-range order or algebraic decay with critical index =1/2, i.e.T=0 is thus a critical point.Work performed within the research program of the Sonder forschungsbereich 125 Aachen-Jülich-Köln     

Influence of the line stiffness on the critical properties of 2d domain walls and polymer systems

A recently developed efficient Monte-Carlo method is used to calculate the critical equilibrium properties of a 2-dimensional system of thermal loops (loop gas) in dependence of the line stiffness energys. With increasing s the critical temperatureT _c (defining an Ising-like behaviour fors<1)decreases monotonically toT _c =0 ats=1 (in units of the line energy). Fors>1,T _c increases monotonically withs and defines anon-universal critical behaviour. The critical line is calculated in a phase diagram (i) as aT _c -versus-s plot showing a dipT _c =0 ats=1 and (ii) in a concentration (c)-versus-s diagram, describing, alternatively, a dilute system of rough polymers. In the latter diagram the critical concentration decreases monotonically withs fors<1 and increases withs fors>1.     

Das kritische magnetische Verhalten von EuO

Mössbauer effect and magnetization measurements were employed in order to study the static and especially the dynamic magnetic properties of the nearly Heisenberg ferromagnet EuO near its Curie temperature,T _c=69.2 °K. The critical exponent β of the spontaneous magnetization was determined to be β=0.34±0.02. It was shown that critical slowing down of spin fluctuations takes place nearT _c with spin relaxation times between 7×10^?11 sec (T=1.01T _c) and 1.5×10^?1 sec (T=1.03T _c). The experimental values of the relaxation time were found to be in satisfactory agreement with theoretically computed ones. Just belowT _c the Mössbauer spectra exhibit relaxation effects, which are characteristic for the occurence of critical super-paramagnetism. Investigations of several samples indicated quantitatively, that critical superparamagnetism has its origin in the non ideal composition of the real crystal.     

Directed-site percolation clusters: The scaling function in dimensions two to six

Exact series for lattices of dimension between 2 and 6 are used to report on the asymptotic features of the scaling function for the average number of clusters in directed percolation, close to, and away from, the most recent estimated intervals forp _c . Scanning of the noncritical regions yields exponent ranges compatible with the undirected percolation equivalents. Close top _c the scaling function varies fairly linearly in terms of the variablez=(p–p _c )s and this result is rather stable particularly bearing in mind the modestly available precision forp _c in higher dimensionalities.     

Distribution of pseudo-critical temperatures and lack of self-averaging in disordered Poland-Scheraga models with different loop exponents

According to recent progresses in the finite size scaling theory of disordered systems, thermodynamic observables are not self-averaging at critical points when the disorder is relevant in the Harris criterion sense. This lack of self-averageness at criticality is directly related to the distribution of pseudo-critical temperatures T_c(i,L) over the ensemble of samples (i) of size L. In this paper, we apply this analysis to disordered Poland-Scheraga models with different loop exponents c, corresponding to marginal and relevant disorder. In all cases, we numerically obtain a Gaussian histogram of pseudo-critical temperatures T_c(i,L) with mean T_c^av(L) and width ΔT_c(L). For the marginal case c=1.5 corresponding to two-dimensional wetting, both the width ΔT_c(L) and the shift [T_c(∞)-T_c^av(L)] decay as L^-1/2, so the exponent is unchanged (ν_random=2=ν_pure) but disorder is relevant and leads to non self-averaging at criticality. For relevant disorder c=1.75, the width ΔT_c(L) and the shift [T_c(∞)-T_c^av(L)] decay with the same new exponent L^-1/νrandom (where ν_random ∼2.7 > 2 > ν_pure) and there is again no self-averaging at criticality. Finally for the value c=2.15, of interest in the context of DNA denaturation, the transition is first-order in the pure case. In the presence of disorder, the width ΔT_c(L) ∼L^-1/2 dominates over the shift [T_c(∞)-T_c^av(L)] ∼L^-1, i.e. there are two correlation length exponents ν=2 and that govern respectively the averaged/typical loop distribution.     

Similarity laws for the density profile of percolation clusters and lattice animals

Analysis of Monte Carlo data shows the density profiles of critical percolation clusters (p=p _c ) to be similar to each other. The same is true for random animals (p=0). The exponents for these scaling laws are determined and agree with those expected from the cluster radii.     

2D-to-3D percolation crossover of metal-insulator composites

Percolation properties and d.c. conductivity were determined for an L²×h-random resistor network model of metal-insulator composite films. The effects of the thickness h on the percolation threshold and conductivity were studied numerically in the limit of an infinite size of the L²-plane parallel to the film. For thicknesses ranging from h/L=0.01 to h/L=0.24, a crossover between a finite-size regime and a saturation regime was observed at h/L≈0.1. In the finite-size regime (h/L?0.01), the percolation threshold scales as pc(h)−pc3∝h−1/x, the exponent x being compatible with that of the critical exponent of the 3D correlation length, ν₃. The conductivity exponent t appeared to depend linearly on the ratio h/L with a slope νD compatible with 2+ν₂, where ν₂ is the 2D correlation length exponent. In the saturation regime, a scaling correction for the percolation threshold was found with an exponent 1+1/ν₃. In this regime we also observed a logarithmic dependence of the conductivity exponent on h/L.     

Time distribution and loss of scaling in granular flow

Two cellular automata models with directed mass flow and internal time scales are studied by numerical simulations. Relaxation rules are a combination of probabilistic critical height (probability of toppling p) and deterministic critical slope processes with internal correlation time t_c equal to the avalanche lifetime, in model A, and ,in model B. In both cases nonuniversal scaling properties of avalanche distributions are found for , where is related to directed percolation threshold in d=3. Distributions of avalanche durations for are studied in detail, exhibiting multifractal scaling behavior in model A, and finite size scaling behavior in model B, and scaling exponents are determined as a function of p. At a phase transition to noncritical steady state occurs. Due to difference in the relaxation mechanisms, avalanche statistics at approaches the parity conserving universality class in model A, and the mean-field universality class in model B. We also estimate roughness exponent at the transition. Received: 29 May 1998 / Revised: 8 September 1998 / Accepted: 10 September 1998     

The field theory approach to percolation processes

We review the field theory approach to percolation processes. Specifically, we focus on the so-called simple and general epidemic processes that display continuous non-equilibrium active to absorbing state phase transitions whose asymptotic features are governed, respectively, by the directed (DP) and dynamic isotropic percolation (dIP) universality classes. We discuss the construction of a field theory representation for these Markovian stochastic processes based on fundamental phenomenological considerations, as well as from a specific microscopic reaction-diffusion model realization. Subsequently we explain how dynamic renormalization group (RG) methods can be applied to obtain the universal properties near the critical point in an expansion about the upper critical dimensions d_c = 4 (DP) and 6 (dIP). We provide a detailed overview of results for critical exponents, scaling functions, crossover phenomena, finite-size scaling, and also briefly comment on the influence of long-range spreading, the presence of a boundary, multispecies generalizations, coupling of the order parameter to other conserved modes, and quenched disorder.     

Random walks in a simple percolation model with two jump frequencies

We consider transport properties of the system in which the good-conducting bonds lie in parallel planes linked by poor-conducting bonds and the concentration p of good-conducting bonds is close to the two-dimensional percolation threshold p_c. The diffusion coefficient D(τ) which describes the random walking in directions along the planes is calculated as a function of variable τ = p − p_c. For τ → 0 the asymptotic relation D(τ)/D(0) − 1 | ∼ |τ|^α is found w α = 2ν − s. Here s is the superconductivity exponent and ν is the correlation length exponent. It is argued that such behavior is to be expected also for more general models.     

具L-近邻键的一维渗流模型临界指数的精确解

应用实空间重正群方法于一维具L-近邻键的点-渗流及键-渗流模型,得到类-温度标度率与类-场标度率,再利用普适标度律得到全部临界指数的精确结果。对于点-渗流模型有α_p=2-L,β_p=0,γ_p=L,δ_p=∞,η_p=1及v_p=L。这与生成函数方法结果一致;对于键-渗流模型有α_p=2-(L(L+1))/2,β_p=0,γ_p=(L(L+1))/2,δ_p=∞,η_p=1及v_p=(L(L+1))/2,其中的“热”临界指数与转移矩阵方法结果一致,磁临界指数是新的结果。由点-渗流及键-渗流模型求出Suzuki的弱普适律的重正化临界指数为φ≡(2-α)/v=1,β≡β/v=0,γ≡γ/v=1,δ≡δ=∞及η≡η=1。即重正化的临界指数不仅与L无关,而且也不依赖于是点抑键的渗流模型,即普适律对Suzuki重正化临界指数仍得以保持。 关键词：     

Phase transition in Ising, XY and Heisenberg magnetic films

The phase transition and magnetic properties of a ferromagnet spin-S, a disordered diluted thin and semi-infinite film with a face-centered cubic lattice are investigated using the high-temperature series expansions technique extrapolated with Padé approximants method for Heisenberg, XY and Ising models. The reduced critical temperature of the system τ_c is studied as function of the thickness of the thin film and the exchange interactions in the bulk, and within the surfaces J_b, J_s and J_⊥, respectively. It is found that τ_c increases with the exchange interactions of surface. The magnetic phase diagrams (τ_c versus the dilution x) and the percolation threshold are obtained. The shifts of the critical temperatures T_c(l) from the bulk value (T_c(∞)/T_c(l) − 1) can be described by a power law l^−λ, where λ = 1/υ is the inverse of the correlation length exponent.