Critical behavior of anisotropic cubic systems with long- and short-range interactions

Critical phenomena in anisotropic cubic N-component spin systems with long- and short-range interactions are investigated and discussed in the regions of weakly long-range, intermediate-range, range, and the long-range potentials. The expressions for the eigenvalues and the critical exponents (n,γ and crossover exponents) in these three regions are derived and their stability is discussed. These results of the systems are compared with those of the same isotropic systems.     

The Crossover Region Between Long-Range and Short-Range Interactions for the Critical Exponents

It is well know that systems with an interaction decaying as a power of the distance may have critical exponents that are different from those of short-range systems. The boundary between long-range and short-range is known, however the behavior in the crossover region is not well understood. In this paper we propose a general form for the crossover function and we compute it in a particular limit. We compare our predictions with the results of numerical simulations for two-dimensional long-range percolation.     

Lévy-flight spreading of epidemic processes leading to percolating clusters

We consider two stochastic processes, the Gribov process and the general epidemic process, that describe the spreading of an infectious disease. In contrast to the usually assumed case of short-range infections that lead, at the critical point, to directed and isotropic percolation respectively, we consider long-range infections with a probability distribution decaying in d dimensions with the distance as . By means of Wilson's momentum shell renormalization-group recursion relations, the critical exponents characterizing the growing fractal clusters are calculated to first order in an -expansion. It is shown that the long-range critical behavior changes continuously to its short-range counterpart for a decay exponent of the infection . Received: 17 July 1998 / Revised: 20 July 1998 / Accepted: 28 July 1998     

Exact results for the Kardar-Parisi-Zhang equation with spatially correlated noise

We investigate the Kardar-Parisi-Zhang (KPZ) equation in d spatial dimensions with Gaussian spatially long-range correlated noise -- characterized by its second moment -- by means of dynamic field theory and the renormalization group. Using a stochastic Cole-Hopf transformation we derive exact exponents and scaling functions for the roughening transition and the smooth phase above the lower critical dimension . Below the lower critical dimension, there is a line marking the stability boundary between the short-range and long-range noise fixed points. For , the general structure of the renormalization-group equations fixes the values of the dynamic and roughness exponents exactly, whereas above , one has to rely on some perturbational techniques. We discuss the location of this stability boundary in light of the exact results derived in this paper, and from results known in the literature. In particular, we conjecture that there might be two qualitatively different strong-coupling phases above and below the lower critical dimension, respectively. Received 5 August 1998     

Second-order phase transition with coexistence of short-range and long-range ferromagnetic interactions in Ba1.7La0.3FeMoO6 compound

In this work, the magnetic properties and critical behavior around ferromagnetic–paramagnetic (FM–PM) phase transition in Ba_1.7La_0.3FeMoO₆ compound have been investigated in detail. This compound exhibits a second-order magnetic phase transition with Curie temperature T_C = 345 K. The critical exponents β, γ, and δ that are determined by using the modified Arrott plots (MAP), the Kouvel–Fisher (KF) and the critical isotherm analysis agree very well. Our results indicate a coexistence of short-range and long-range ferromagnetic (FM) interactions in Ba_1.7La_0.3FeMoO₆ compound. The existence of long-range FM interactions in this compound can be associated with the crystal structure of materials with long-range Fe/Mo ordering parameter and strength of double-exchange interaction, whereas the existence of the short-range FM interactions can be explained by magnetic inhomogeneity and FM clusters.     

Cross-Over Between First-Order and Critical Wetting at the Liquid-Vapour Interface of n-Alkane/Methanol Mixtures: Tricritical Wetting and Critical Prewetting

A simple mean-field theory is presented which describes the basic observations of recent experiments revealing rich wetting behaviour of n-alkane/methanol mixtures at the liquid-vapour interface. The theory, qualitative and in part heuristic, is based on a microscopic lattice-gas model from which a Cahn–Landau approach is distilled. Besides the physics associated with the short-range components of the intermolecular interactions, effects of the long-range tails of the net van der Waals forces between interfaces are also taken into account. Further, gravitational thinning of the wetting phase is incorporated. The calculation of the spreading coefficient S is extended to the experimentally relevant situation in which the bulk adsorbate is slightly away from two-phase coexistence due to gravity. Analysis of this novel approximation to S for systems with short-range forces leads to the conclusion that the surface specific heat exponents _s =1,1/2, and 0, for first-order wetting, tricritical wetting and critical wetting, respectively, are robust with respect to (weak) gravitational thinning, consistently with experiment. For three different systems the adsorption is calculated as a function of temperature and compared with the experimentally measured ellipticity. Including weak long-range forces which favour wetting in the theory does not visibly alter the critical wetting transition for the nonane/methanol mixture, in contrast with the generic expectation of first-order wetting for such systems, but in good agreement with experiment. For decane/methanol weak long-range forces bring the transition very close to the prewetting critical point, leading to an adsorption behaviour closely reminiscent of short-range tricritical wetting, observed experimentally for alkane chain length between 9.6 and 10. Finally, for undecane/methanol the transition is clearly of first order. First-order wetting is also seen in the experiment.     

斯格明子相关的螺旋磁有序体系的临界行为

介绍了与斯格明子相关的螺旋磁有序体系的临界行为.首先阐述了连续相变中的临界现象、临界指数、标度律、普适性等概念;随后介绍了磁相变体系中几种临界指数的获得方法,包括直流磁性迭代法、磁熵变法;进而,分析了几类与斯格明子相关的螺旋磁有序体系的临界行为.MnSi是典型的斯格明子材料,临界指数显示其磁性行为符合三重临界行为.MnSi的临界行为揭示:外磁场可以抑制这一体系在零场下的一级相变,使其转变为二级相变,从而在螺旋磁有序、锥形磁有序、顺磁相的三相交汇点形成三重临界点.斯格明子体系FeGe和Cu_2OSeO_3的临界行为符合三维海森伯相互作用,表明它们的磁性行为主要是由近邻的各向同性的自旋耦合作用所决定;而Fe_(1-x)Co_xSi和新发现的斯格明子体系Fe_(1.5-x)Co_xRh_(0.5)MoN的临界行为显示Co掺杂可以有效地调制其中的磁性耦合.对螺旋磁有序体系的临界行为研究表明,尽管这些体系都表现出类似的斯格明子态,但是它们的磁性耦合机制却大不相同,并且其耦合机制可以受到外界手段的调制.最后,根据普适性原理和标度方程,阐述了一种构建磁场诱导相变体系在临界温度附近H-T相图的方法.     

A model for anomalous directed percolation

We introduce a model for the spreading of epidemics by long-range infections and investigate the critical behaviour at the spreading transition. The model generalizes directed bond percolation and is characterized by a probability distribution for long-range infections which decays in d spatial dimensions as . Extensive numerical simulations are performed in order to determine the density exponent and the correlation length exponents and for various values of . We observe that these exponents vary continuously with , in agreement with recent field-theoretic predictions. We also study a model for pairwise annihilation of particles with algebraically distributed long-range interactions. Received: 4 September 1998 / Accepted: 22 September 1998     

Exact entanglement studies of strongly correlated systems: role of long-range interactions and symmetries of the system

We study the bipartite entanglement of strongly correlated systems using exact diagonalization techniques. In particular, we examine how the entanglement changes in the presence of long-range interactions by studying the Pariser-Parr-Pople model with long-range interactions. We compare the results for this model with those obtained for the Hubbard and Heisenberg models with short-range interactions. This study helps us to understand why the density matrix renormalization group (DMRG) technique is so successful even in the presence of long-range interactions. To better understand the behavior of long-range interactions and why the DMRG works well with it, we study the entanglement spectrum of the ground state and a few excited states of finite chains. We also investigate if the symmetry properties of a state vector have any significance in relation to its entanglement. Finally, we make an interesting observation on the entanglement profiles of different states (across the energy spectrum) in comparison with the corresponding profile of the density of states. We use isotropic chains and a molecule with non-Abelian symmetry for these numerical investigations.     

Dynamic transitions in Domany-Kinzel cellular automata on small-world network

We study Domany-Kinzel cellular automata on small-world network. Every link on a one dimensional chain is rewired and coupled with any node with probability p. We observe that, the introduction of long-range interactions does not remove the critical character of the model and the system still exhibits a well-defined phase transition to absorbing state. In case of directed percolation (DP), we observe a very anomalous behavior as a function of size. The system shows long lived metastable states and a jump in order parameter. This jump vanishes in thermodynamic limit and we recover second-order transition. The critical exponents are not equal to the mean-field values even for large p. However, for compact directed percolation(CDP), the critical exponents reach their mean-field values even for small p.     

长程作用下Gauss系统的临界温度

利用傅里叶变换的方法，严格求解了d维(d=1，2和3)超立方晶格和二维三角晶格上具有长程 相互作用的Gauss模型(这里考虑的长程作用有幂指数、指数和对数三种形式).得到了这些情 况下系统的临界点(温度)，并对不同形式的长程作用对临界点的影响进行了比较.结果表明 ，长程相互作用的存在，使得系统的临界温度有了一定程度的升高，它们对系统临界温度的 影响与其衰减的快慢有关. 关键词： Gauss模型 临界点 超立方晶格 三角晶格     

Critical Behavior of Gaussian Model on X Fractal Lattices in External Magnetic Fields

Using the renormalization group method, the critical behavior of Gaussian model is studied in external magnetic fields on X fractal lattices embedded in two-dimensional and d-dimensional (d ＞ 2) Euclidean spaces, respectively. Critical points and exponents are calculated. It is found that there is long-range order at finite temperature for this model, and that the critical points do not change with the space dimensionality d (or the fractal dimensionality dr). It is also found that the critical exponents are very different from results of Ising model on the same lattices, and that the exponents on X lattices are different from the exact results on translationally symmetric lattices.     

Kinetic Gaussian Model with Long-Range Interactions

In this paper dynamical critical phenomena of the Gaussian model with long-range interactions decaying as 1/r^d+δ (δ>0) on d-dimensional hypercubic lattices (d=1, 2, and 3) are studied. First, the critical points are exactly calculated, and it is found that the critical points depend on the value of δ and the range of interactions. Then the critical dynamics is considered. We calculate the time evolutions of the local magnetizations and the spin-spin correlation functions, and further the dynamic critical exponents are obtained. For one-, two- and three-dimensional lattices, it is found that the dynamic critical exponents are all z=2 if δ>2, which agrees with the result when only considering nearest neighboring interactions, and that they are all δ if 0<δ<2. It shows that the dynamic critical exponents are independent of the spatial dimensionality but depend on the value of δ.     

The Kauffman model on small-world topology

We apply Kauffman's automata on small-world networks to study the crossover between the short-range and the infinite-range case. We perform accurate calculations on square lattices to obtain both critical exponents and fractal dimensions. Particularly, we find an increase of the damage propagation and a decrease in the fractal dimensions when adding long-range connections.     

Lattice dynamics of II-VI mixed semiconductor ZnS1−xSex

Lattice dynamical properties of II-VI compounds having zinc-blende structure have been calculated by three-body shell model. This model incorporates the effect of the short-range repulsive interactions up to and including the second nearest neighbours, in addition to the long-range Coulombic interactions in the frame work of the rigid-shell model with both the ions are polarizable. The model involves in total eleven disposable parameters. Using the above proposed model the phonon dispersion relations for mixed II-VI semiconductor ZnS_1−xSe_x are plotted. We find an overall good agreement with the experimental results. The application of the present model has been made to calculate the phonon dispersion relations of ZnS, ZnSe and mixed semiconductor ZnS_1−xSe_x. The comparison of the theoretical results with the available experimental has been made along high symmetry directions. A reasonably good agreement is observed between theory and experiments.     

Renormalization group treatment of the scaling properties of finite systems with subleading long-range interaction

The finite size behavior of the susceptibility, Binder cumulant and some even moments of the magnetization of a fully finite O(n) cubic system of size L are analyzed and the corresponding scaling functions are derived within a field-theoretic ɛ-expansion scheme under periodic boundary conditions. We suppose a van der Waals type long-range interaction falling apart with the distance r as r ^{- (d + σ)}, where 2 < σ < 4, which does not change the short-range critical exponents of the system. Despite that the system belongs to the short-range universality class it is shown that above the bulk critical temperature T _c the finite-size corrections decay in a power-in-L, and not in an exponential-in-L law, which is normally believed to be a characteristic feature for such systems. Received 8 August 2001     

Long-range connections, quantum magnets and dilute contact processes

In this article, we briefly review the critical behaviour of a long-range percolation model in which any two sites are connected with a probability that falls off algebraically with the distance. The results of this percolation transition are used to describe the quantum phase transitions in a dilute transverse Ising model at the percolation threshold p_c of the long-range connected lattice. In the similar spirit, we propose a new model of a contact process defined on the same long-range diluted lattice and explore the transitions at p_c. The long-range nature of the percolation transition allows us to evaluate some critical exponents exactly in both the above models. Moreover, mean field theory is valid for a wide region of parameter space. In either case, the strength of Griffiths McCoy singularities are tunable as the range parameter is varied.     

A damage model based on failure threshold weakening

A variety of studies have modeled the physics of material deformation and damage as examples of generalized phase transitions, involving either critical phenomena or spinodal nucleation. Here we study a model for frictional sliding with long-range interactions and recurrent damage that is parameterized by a process of damage and partial healing during sliding. We introduce a failure threshold weakening parameter into the cellular automaton slider-block model which allows blocks to fail at a reduced failure threshold for all subsequent failures during an event. We show that a critical point is reached beyond which the probability of a system-wide event scales with this weakening parameter. We provide a mapping to the percolation transition, and show that the values of the scaling exponents approach the values for mean-field percolation (spinodal nucleation) as lattice size L is increased for fixed R. We also examine the effect of the weakening parameter on the frequency-magnitude scaling relationship and the ergodic behavior of the model.     

Scaling of the interface roughness in Fe-Cr superlattices: self-affine versus non-self-affine

We have analyzed kinetic roughening in Fe-Cr superlattices by energy-filtered transmission electron microscopy. The direct access to individual interfaces provides both static and dynamic roughness exponents. We find an anomalous non-self-affine scaling of the interface roughness with a time dependent local roughness at short length scales. While the deposition conditions affect strongly the long-range dynamics, the anomalous short-range exponent remains unchanged. The different short- and long-range dynamics outline the importance of long-range interactions in kinetic roughening.     

Phase diagram of the two-dimensional ferromagnetic three-color Ashkin-Teller model

In the present work we study the critical properties of the ferromagnetic three-color Ashkin-Teller model (3AT) by means of a Migdal-Kadanoff renormalization group approach on a diamond-like hierarchical lattice. The analysis of the fixed points and flux diagram of the recursion relations is used to determine the corresponding phase diagram (including its symmetry properties) and critical exponents. Our numerical results show the presence of four universality classes, three of them are associated to the Potts model with q=2, 4 and 6 states. Finally, a connection between our findings and some known results from the literature is presented.