An exact one-particle-irreducible renormalization-group generator for critical phenomena

An exact one-particle-irreducible renormalization-group generator for critical phenomena is derived by an infinitesimal saddle-point expansion. This replaces the usual field-theoretic loop-expansion for the free energy and Green's functions with an explicit differential equation.     

Introduction to real-space renormalization-group methods in critical and chaotic phenomena

The methods of the real-space renormalization group, and their application to critical and chaotic phenomena are reviewed. The article consists of two parts: the first part deals with phase transitions and critical phenomena; the second part, bifurcations and transitions to chaos. We begin with an introduction to the phenomenology of phase transitions and critical phenomena. Seminal concepts such as scaling and universality, and their characterization by critical exponents are discussed. The basic ideas of the renormalization group are then explained. A survey of real-space renormalization-group methods: decimation, Migdal-Kadanoff approximation, cumulant and cluster expansions, is given. The Hamiltonian formulation of classical statistical systems into quantum mechanical systems by the method of the transfer matrix is introduced. Quantum renormalization-group methods of truncation and projection, and their application to the transcribed quantum mechanical Ising model in a transverse field are illustrated. Finally, the quantum cumulant-expansion method as applied to the one-dimensional quantum mechanical XY model is discussed. The second part of the article is devoted to the subject of bifurcations and transitions to chaos. The three most commonly discussed kinds of bifurcations: the pitchfork, tangent and Hopf bifurcations, and the associated routes to chaos: period doubling, intermittency and quasiperiodicity are discussed. Period doubling based on the logistic map is explained in detail. Universality and its expression in terms of functional renormalization-group equations is discussed. The Liapunov characteristic exponent and its analogy to the order parameter are introduced. The effect of external noise and its universal scaling feature are shown. The simplest characterizations of the Hénon strange attractor are intuitively illustrated. The purpose of this article is primarily pedagogical. The similarity between critical and chaotic phenomena is a recurrent theme that underlines the importance and usefulness of such concepts as scaling, renormalization and universality.     

Critical thermodynamics of two-dimensional systems in the five-loop renormalization-group approximation

The paper is devoted to the calculation of renormalization-group (RG) functions in the O(n)-symmetry two-dimensional model of the λ? ⁴ type in the five-loop approximation and to an analysis of the critical behavior of systems described by this model. Five-loop expansions for the β function and the critical indices are determined in bulk theory. They are summed up using the Padé-Borel and Padé-Borel-Le Roy methods, making it possible to optimize the summation procedure and to estimate the accuracy of the obtained numerical values. It is shown that in the Ising (n=1) case, as well as in other cases, the inclusion of the five-loop contribution to the β function displaces the coordinate of the Wilson fixed point only insignificantly, leaving it outside the interval formed by the results of computations on lattices; even “spreads” of the error in the renormalization group and lattice estimates do not overlap. This discrepancy is attributed to the effect of the nonanalytic com-ponent of the β function, which cannot be determined in perturbation theory. A computation of critical indices proves that, although the inclusion of the five-loop terms in the corresponding RG expansion slightly improves the concordance with the exact results, the nonanalytic contributions are apparently also significant in this case.     

Critical phenomena in antiferroelectrics

An equation of state for an antiferroelectric is proposed, which couples the field strength, temperature, and polarization, and has isotherms that display double hysteresis loops and a critical point. On the phase diagram of field versus temperature two critical points, symmetrically placed with respect to the temperature axis, are observed. The presence of a critical point on the temperature-polarization and temperature-entropy diagrams is also shown.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 27–31, September, 1972.The author wishes to thank Prof. Semenchenko for his interest in this work and valuable comments.     

Critical phenomena and relativistic gravitational collapse

Recent calculations have shown the existence of critical phenomena in general relativity associated with the collapse of wavepackets of massless fields that are near, in parameter space, the onset of black hole formation (the critical point). Two physically distinct systems have been explored: collapse of spherically-symmetric massless scalar field and collapse of vacuum, axisymmetric gravitational waves. Nonlinear effects dominate near the critical point. Black-hole mass serves as an order parameter and has a power-law dependence on critical separation in the supercritical region of parameter space. Remarkably, the values of the critical exponent of the power law are nearly identical in the two systems. The nonlinearity induces the fields to oscillate. Each successive oscillation is an echo, obeying a spatial and temporal scaling relation.     

Critical phenomena at edges and corners

Using Monte-Carlo techniques, the critical behaviour at edges and corners of the three-dimensional Ising model is studied. In particular, the critical exponent of the local magnetization at edges formed by two intersecting free surfaces is estimated to be, as a function of the opening angle , for , for , and for . The critical exponent of the corner magnetization of a cube is found to be . The Monte-Carlo estimates are compared to results of mean field theory, renormalization group calculations and high temperature series expansions. Received: 29 January 1998 / Accepted: 17 March 1998     

Critical phenomena in FINEMET alloy

The Fe_73.5Cu₁Nb₃Si_13.5B₉ FINEMET alloy has been prepared by the rapid solidification technique. The critical behaviour of this alloy in the amorphous as well as in its nanocrystalline states has been studied near their respective Curie temperatures. From the values of the critical exponents one can conclude that the alloy behaves like a 3D Heisenberg ferromagnet in the amorphous and nanocrystalline states. But there exists a slight increase in the value of β for the alloy annealed at 823 K (the nanocrystalline state) as observed in most of the amorphous alloys.     

非晶物质中的临界现象

非晶态材料有着复杂的原子结构(短程有序、长程无序)和特殊的物理性质,其临界现象和相变问题一直受到学术界关注.非晶合金,又称为金属玻璃,是一种新型的非晶态材料,具有很高的强度和优异的弹性.从微观的角度来看,非晶合金可以看作是一个多粒子系统.临界现象的研究对认识和理解多粒子系统之间的相互作用有深刻的意义.本文主要讨论非晶合金中的临界现象,包括非晶合金从制备过程、微观结构到宏观的力学性能以及磁性方面存在的临界现象,并分析这些临界现象之间的内在联系,进而深入理解非晶合金的微观结构对其宏观性质的影响.这为认识非晶合金的形成本质,提高服役可靠性,探索具有实际应用价值的非晶合金提供理论依据.     

Critical phenomena in ferromagnetic superlattices

Within the framework of the high-temperature series expansions technique, we examine the phase transition and the critical phenomena of a two-component superlattice with simple cubic structure, through three models: Ising, XY and Heisenberg. The reduced critical temperature of the system is studied as a function of the thickness of the constituents and the exchange interactions in each material, and within the interface. We show the existence of a critical thickness of the unit cell at which the reduced critical temperature of the binary superlattice remains insensitive to the exchange coupling within the interfaces. The values of the effective critical exponent γ _eff associated with the magnetic susceptibility agreed with the universal classes in the limit cases where the superlattice is still comparable to an infinite simple cubic lattice. We attribute the breakdown in the universality hypothesis to the crossover effects.     

Critical phenomena at perfect and non-perfect surfaces

The effect of imperfections on surface critical properties is studied for Ising models with nearest-neighbour ferromagnetic couplings on simple cubic lattices. In particular, results of Monte Carlo simulations for flat, perfect surfaces are compared to those for flat surfaces with random, “weak” or “strong”, interactions between neighbouring spins in the surface layer, and for surfaces with steps of monoatomic height. Surface critical exponents at the ordinary transition, in particular ,are found to be robust against these perturbations. Received: 7 October 1997 / Accepted: 19 November 1997     

Critical phenomena studied via nuclear techniques

Since pioneering NMR studies of magnetic critical phenomena in MnF₂ by Heller, a range of hyperfine techniques, particularly Mössbauer effect and perturbed angular correlations, have been successfully applied to the problem. When optimally designed, these methods provide experimental approaches that are in some ways more successful than magnetic neutron scattering. Although the number of hyperfine results now number in the hundreds, only a few are sufficiently asymptotic to provide clearcut tests of theory. In this paper we outline the theoretical framework used to describe the modern theory of critical phenomena, discuss the strengths and weakness of various experimental methods, and provide a selection of the best experiments that test the theory. Whereas we emphasize hyperfine interaction studies, we also describe the principles and results of other methods where appropriate. Particular topics include: (1) static critical phenomena in chemically ordered systems studied as a function ofn andd, the spin and lattice dimensionality; (2) static critical phenomena in magnetically disordered systems studied as a function of the type of disorder; and (3) dynamic critical phenomena in chemically ordered systems.     

Critical phenomena of ferromagnetic cobalt disulfide

The simultaneous measurements of magnetization and resistivity of a ferromagnetic CoS₂ near the critical temperature have shown that the negative divergence of $\text{d}\text{ρ}\text{d}\text{T}$ occurs exactly at the Curie temperature of 115.9 K. The magnetization satisfies the static scaling law.     

Critical phenomena in discrete-time interconnection networks

We present theoretical and numerical results for the performance of a multiprocessor network modeled as a ring and as a toroidal square lattice of nodes with local processors that generate messages for output ports/buffers. The output buffers are assumed to have infinite capacity, and the service time is deterministic. Two models are considered. One assumes that every processor generates messages with rate λ per time slot and per output port/buffer. The other model considers that the generation rate of a node depends on the intensity of the flow of arriving messages. Explicit expressions for the distribution of queue lengths, the average number of messages in the buffers, the average latency, and the critical network load depending on the distance between the source and the destination are obtained. Simulation results show excellent agreement with theoretical predictions based on the assumption of independent queues.     

Critical phenomena in nonequilibrium phase transitions

We discuss a number of models associated with phase transitions in purely kinetic models where detailed balance does not hold as in thermal equilibrium systems. These models include some of the features of heterogeneous catalysis on a surface, and are used to examine the effect of local correlations on the reaction process. We argue that many models which do not include desorption will show the same kind of critical behavior if they have a continuous transition to a poisoned state. We also present results for a model with a continuous transition to a non-poisoned state. The effect of adding desorption, diffusion, and anisotropy to these idealized models is also discussed.     

Critical phenomena in model neural networks

Dynamical properties of a sparsely connected model network composed of N binary decision elements with randomly chosen asymmetrical couplings are studied statistically. In the thermodynamic limit exact results based on the time evolution of the Hamming distance of two different initial states predict the occurrence of two phases upon varying any of the network parameters: a frozen and a chaotic phase. Extensive computer simulations agree well with the results. The model is also shown to share behavioral similarities with the Kauffmann model pointing to the existence of formal relationships.