Critical Behavior Of System Research Articles

Effects of instability of quantum critical behaviour in systems with quenched impurities

The one-loop renormalization group equations are derived and analysed for a wide class of quantum statistical models describing the critical behaviour of systems with short-range correlated random impurities. The classical and quantum critical properties are considered to first order in a double ε-expansion. It is proven that the quantum critical behaviour of disordered systems with short-range random correlation is equivalent to the classical critical behaviour of certain systems with extended random impurities. The reliability of the double ε-expansion method and the interpretation of the results are discussed.

Surface critical behavior in systems with nonequilibrium phase transitions

We study the surface critical behavior of branching-annihilating random walks with an even number of offspring (BARW) and directed percolation (DP) using a variety of theoretical techniques. Above the upper critical dimensions d(c), with d(c)=4 (DP) and d(c)=2 (BARW), we use mean field-theory to analyze the surface phase diagrams using the standard classification into ordinary, special, surface, and extraordinary transitions. For the case of BARW, at or below the upper critical dimension d</=d(c), we use field theoretic methods to study the effects of fluctuations. As in the bulk, the field-theory suffers from technical difficulties associated with the presence of a second critical dimension. However, we are still able to analyze the phase diagrams for BARW in d=1 and 2, which turn out to be very different from their mean field analog. Furthermore, for the case of BARW only (and not for DP), we find two independent surface beta(1) exponents in d=1, arising from two distinct definitions of the order parameter. Using an exact duality transformation on a lattice BARW model in d=1, we uncover a relationship between these two surface beta(1) exponents at the ordinary and special transitions. Many of our predictions are supported using Monte Carlo simulations of two different models belonging to the BARW universality class.

Critical behavior in systems with a long-range correlated frozen-in random field

The critical behavior of inhomogeneous systems with a frozen-in random field having a nonlocal correlation function decaying according to the power law $|\mathbf{x}\ensuremath{-}{\mathbf{x}}^{\ensuremath{'}}{|}^{\ensuremath{-}b}$ is considered. The problem is studied within the context of a model which partially considers interactions of fluctuations and belongs in the same universality class as the spherical model. Depending on the relationship between the power b and the space dimensionality d, a new critical behavior arises.

Static and dynamic critical properties of 3D systems with long-range correlated quenched defects

A field-theoretic description of static and purely relaxational dynamic critical behaviour of systems with quenched defects obeying power law correlations ~|x|-a for large separations x is given. Directly, for three-dimensional systems and for different values of the correlation parameter, 2 a3, a renormalization analysis of the scaling functions in the two-loop approximation is carried out, and the fixed points corresponding to the stability of various types of critical behaviour are identified. The obtained results essentially differ from results evaluated by a double ,-expansion. The static and dynamic critical exponents in the two-loop approximation are calculated with the use of the Pade-Borel summation technique.

Critical behaviour of 3D systems with long-range correlated quenched defects

A field-theoretic description of the critical behaviour of systems with quenched defects obeying a power law correlations $\sim |{\bf x}|^{-a}$ for large separations ${\bf x}$ is given. Directly for three-dimensional systems and different values of correlation parameter $2\leq a \leq 3$ a renormalization analysis of scaling function in the two-loop approximation is carried out, and the fixed points corresponding to stability of the various types of critical behaviour are identified. The obtained results essentially differ from results evaluated by double $\epsilon, \delta$ - expansion. The critical exponents in the two-loop approximation are calculated with the use of the Pade-Borel summation technique.

Spin-glass effects on critical behavior in systems with extended quenched disorder

A renormalization-group approach is used for investigating the effects of the Parisi replica symmetry breaking (``spin-glass effects'') on critical properties of d-dimensional systems with extended quenched disorder along ${\ensuremath{\varepsilon}}_{d}$ directions. Within a double expansion scheme to first order in $(\ensuremath{\varepsilon}=4\ensuremath{-}d, {\ensuremath{\varepsilon}}_{d})$ it is found that for $d&lt;~4+{\ensuremath{\varepsilon}}_{d}$ and order-parameter dimensionalities $n&lt;~4,$ a full fixed-point instability with a consequent runaway towards a strong-coupling regime occurs in the parameter space while, for $d&lt;4$ and $n&gt;4,$ compatibly with the modified Harris criterion, an additional marginally stable fixed point exists which might control a new anisotropic random critical behavior. In any case, the traditional replica-symmetric Dorogovtsev-like solution for classical and quantum systems is unstable against the replica symmetry-breaking modes.

Influence of long-range interactions on the critical behavior of systems with a negative Fisher exponent

The influence of long-range interactions decaying in d dimensions as ${1/R}^{d+\ensuremath{\sigma}}$ on the critical behavior of systems with negativeFisher's correlation-function exponent for short-range interactions, ${\ensuremath{\eta}}_{\mathrm{SR}}&lt;0,$ is reexamined. Such systems, typically described by ${\ensuremath{\varphi}}^{3}$-field theories, are, e.g., the Potts model in the percolation limit, the Edwards-Anderson spin-glass model, and the Yang-Lee edge singularity. In contrast to preceding studies, it is shown by means of Wilson's momentum-shell renormalization-group recursion relations that the long-range interactions dominate as long as $\ensuremath{\sigma}&lt;2\ensuremath{-}{\ensuremath{\eta}}_{\mathrm{SR}}.$ Exponents change continuously to their short-range values at the boundary of this region.

Surface Critical Behavior in Systems with Absorbing States

We present a general scaling theory for the surface critical behavior of nonequilibrium systems with phase transitions into absorbing states. The theory allows for two independent surface exponents which satisfy generalized hyperscaling relations. As an application we study a generalized version of directed percolation with two absorbing states. We find two distinct surface universality classes associated with inactive and reflective walls. Our results indicate that the exponents associated with these two surface universality classes are closely connected.

Geometrical Universality in Vibrational Dynamics

A good generalization of the Euclidean dimension to disordered systems and noncrystalline structures is commonly required to be related to large scale geometry and it is expected to be independent of local geometrical modifications. The spectral dimension, defined according to the low frequency density of vibrational states, appears to be the best candidate as far as dynamical and thermodynamical properties are concerned. In this letter we give the rigorous analytical proof of its independence of finite scale geometry. We show that the spectral dimension is invariant under local rescaling of couplings and under addition of finite range couplings, or infinite range couplings decaying faster then a characteristic power law. We also prove that it is left unchanged by coarse graining transformations, which are the generalization to graphs and networks of the usual decimation on regular structures. A quite important consequence of all these properties is the possibility of dealing with simplified geometrical models with nearest-neighbors interactions to study the critical behavior of systems with geometrical disorder.

A Numerical Study of the Critical Line of Kauffman Networks

Kauffman networks were introduced in 1969 as a model of genetic regulatory systems. One of the most striking successes of this model is its ability to reproduce, for a critical value of its parameters, the observed scaling laws of the average cell replication time and of the average number of cell types in a given organism vs. the number of genes. Yet, the numerical evidence for such scaling laws in the model is still unsatisfactory, and restricted to a particular critical point, while we expect that the scaling behaviour is universal along the critical line. In this paper we try to sharpen the evidence for the scaling behaviour of critical systems, carrying on a detailed numerical investigation of their properties. We measure the length of the cycles (which in the model represents the period of cell cycles) and their number (which represents the number of cell types) for a point of the critical line different from the only one previously studied. Our results seem to confirm that such quantities scale as √Nfor all critical systems, at least for lengths and numbers small enough. On the other hand, we found that their probability distributions are very broad (power-law like) and become broader with system size. This means that there is an effective scale of the length and of the number of cycles that increases much faster than √N, and in the infinite size limit the biological analogy found by Kauffman may be lost. A numerical study of the modular structure of critical networks supports this conclusion. The implications of this fact for the biological interpretation of the model are briefly discussed. Finally, we found that the typical weight of the attraction basins tends to zero as a power law in the infinite size limit, with an exponent which seems to be universal along the whole critical line.

Infinite numbers of absorbing states: Critical behavior

We briefly review recent progress in elucidating the time-independent critical behavior of systems with infinite numbers of static absorbing states, and show that the critical exponent describing the decay with time of the order parameter right at the critical point is the same as that of the directed percolation problem.

Critical behavior of systems with many absorbing states.

Langevin equations describing the phase transitions of a variety of microscopic models with an infinite number of absorbing states and no additional symmetries are proposed. Renormalization group analysis shows that all of these models have static exponents belonging in the universality class of Reggeon field theory. Connections to epidemic models with immunization are discussed.

Dynamic surface critical behavior of systems with conserved bulk order parameter: Detailed RG analysis of the semi-infinite extensions of modelB with and without nonconservative surface terms

The dynamic surface critical behavior of macroscopic systems whose dynamic bulk critical behavior is described by model $B$ is investigated. The semi-infinite extensions of bulk model $B$ introduced in a previous treatment [Phys. Rev. B 49, 2846 (1994)] and called models $B_A$ and $B_B$ are studied in detail by means of field-theoretic RG methods. The distinctive feature of these models is the presence or absence of relevant nonconservative surface terms. The earlier results on the structure of the required counterterms, the flow equations, and the representation of the surface critical exponents of dynamic quantities at the ordinary and special phase transitions in terms of static bulk and surface exponents are corroborated by means of an explicit two-loop calculation for $4-\epsilon$ dimensions. For parameter values corresponding to a given static surface universality class, models $B_A$ and $B_B$ represent distinct dynamic surface universality classes. Differences between these manifest themselves in the behavior of scaling functions for dynamic surface susceptibilities. RG-improved perturbation theory is used to compute some of these susceptibilities to one-loop order.

Universality classes for the dynamic surface critical behavior of systems with relaxational dynamics.

We investigate the problem of which universality classes of dynamic surface critical behavior exist for semi-infinite systems whose dynamic bulk critical behavior and static surface critical behavior are representative of a given dynamic bulk universality class and a given static surface universality class. To this end systems whose bulk dynamics are described by either model B or model A of Halperin, Hohenberg, and Ma are considered, where the dynamics are allowed to be modified near the surface as follows: In the case of model B, surface terms that locally break the conservation law for the order-parameter density \ensuremath{\varphi} are allowed; in the case of model A, \ensuremath{\varphi} is assumed to be locally conserved near the surface. Semi-infinite field-theory models are constructed that are (i) compatible with the requested bulk dynamics and the requirements of causality, detailed balance, and relaxation to thermal equilibrium and (ii) minimal in the sense that no redundant or irrelevant surface terms are included. The boundary conditions to which the surface terms of the action correspond are worked out. For model B it is shown that nonconservative surface terms are relevant. Their strength can be parametrized by a coupling constant c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$\ensuremath{\ge}0 whose inverse plays the role of an extrapolation length for the auxiliary (Martin-Siggia-Rose) field \ensuremath{\varphi}\ifmmode \tilde{}\else \~{}\fi{}. Thus two distinct semi-infinite extensions of model B exist\char22{}one with c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$g0 called model ${\mathit{B}}_{\mathit{A}}$, and one with c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$=0 called model ${\mathit{B}}_{\mathit{B}}$\char22{}and each static surface universality class splits up into two dynamic surface universality classes.

A new approach to the critical behaviour of systems exhibiting a dimensional crossover

The critical behaviour of systems which exhibit a dimensional crossover is studied. The key new ingredient being an explicitly L dependent renormalization group, L being the characteristic “size” of the system, and a set of renormalization schemes which lead to manifest dimensional reduction in the loop expansion. As a paradigm we study an Ising-like system on S 1×R 3−ϵ, (0≤ϵ≤1). t = T−T c(L) is determined to be the relevant variable rather than T−T c(∞), and a perturbative expansion about the L dependent reduced fixed point performed as opposed to the bulk fixed point. A running coupling constant, effective susceptibility and correlation length exponents are defined and computed to lowest order in perturbation theory. These interpolate between the behaviours expected of an Ising model in 4−ϵ and 3−ϵ dimensions as tL 2→∞( t→0), and tL 2→0, respectively. A scaling law γ eff = ν eff (2− η eff ) is conjectured and shown to hold to lowest order in perturbation theory. The formalism generalizes to higher orders.

Logarithmic effects on the critical behavior of superfluids in random media.

The effects of logarithmic corrections on the critical behavior of systems near a phase transition with marginally irrelevant quenched disorder are examined, specifically in the context of superfluid $^{4}\mathrm{He}$ in porous media. It is argued that these corrections are likely to change the critical behavior of the specific heat to an inverse logarithmic cusp, and alter, to a new universal constant, the magnitude of the amplitude ratio relating the critical specific heat to the superfluid density. An \ensuremath{\varepsilon} expansion is carried out along the line on which the randomness is marginal. The results also have implications for the behavior of dirty high-temperature superconductors.

Self-organized criticality: Goldstone modes and their interactions.

It is shown that critical behavior of systems with self-organized criticality can be explained as a Goldstone-mode phenomenon. The nonlinear interaction of the Goldstone mode causes the nontrivial critical exponents. Two more models are introduced with self-organized criticality features.

Phase equilibria in copolymer/homopolymer ternary blends: Molecular weight effects

The phase behavior of ternary blends of an A–B random copolymer with two homopolymers (A and B) is investigated within the Flory–Huggins lattice theory. We restrict consideration to the formation of (isotropic) liquid phases. For compositionally symmetric systems in which the two homopolymers have equal molecular weights, two different topologies are found for the phase diagrams according to the length of the copolymer relative to the homopolymers. Namely, upon lowering the temperature, a three-liquid-phase region emerges either continuously via a tricritical point if the copolymer is long enough, or discontinuously otherwise. This change in phase behavior, an entropy of mixing effect, occurs when the copolymer length is a fraction 2/5 of the homopolymer molecular weight. The properties of the (symmetric) tricritical points are discussed, as well as the phase behavior of systems in which the copolymer is not symmetrical in composition and/or the two homopolymers differ in size. This Flory–Huggins approach should be valid for blends containing random copolymers, but also, at high enough temperatures, for blends that contain block copolymers, the temperature range of validity being broader for smaller block copolymers. The critical behavior of systems containing block copolymers is described by the simple Flory–Huggins theory over a range delineated by isotropic Lifshitz points. These Lifshitz points are located within the random phase approximation.

SURFACE CRITICAL BEHAVIOUR OF SOME TWO-DIMENSIONAL LATTICE MODELS

A review is given of recent work on the ordinary surface critical behaviour of systems in two dimensions. Several models of interest in statistical mechanics are considered: Potts model, percolation, Ising clusters, ZN-model, O(n) model and polymers. Numerical results for surface exponents, obtained by suitable finite size scaling extrapolations, are discussed in the light of recent advances based on the conformal invariance approach. Surface exponents are often seen as important tests of conformal invariance predictions. In other cases these exponents provide important information for a location of the problem within the classification schemes offered by the conformal approach, and a determination of its universality class. A relevant example of the first aspect is the study of the q-state Potts model with q near 4, for which an analytical study of logarithmic scaling corrections is needed to achieve a successful test. The latter point of view applies, e.g., to the more controversial cases of polymers at the theta point and critical Ising clusters. Emphasis is put on the importance of an integrated study of both bulk and surface properties. Relevant issues, like the possible existence of analytical expressions for the indices in particular model families, or of general relationships between bulk and surface exponents, are critically discussed. The new problem of critical behaviour at fractal boundaries is also considered for random (RW) and self-avoiding walks (SAW). From the numerical analysis of this problem remarkable universalities of the surface exponents seem to emerge, which, in the case of SAW’s, are still far from being understood.

Critical behaviour of systems with cubic anisotropy and random impurities

Renormalisation group recursion relations are obtained to two-loop order for an n-component Landau-Ginzburg-Wilson model containing both hypercubic anisotropy and quenched random impurities. The fixed points are enumerated and their stability properties analysed, using the epsilon expansion about four spatial dimensions to second order. The existence of a new fixed point corresponding to an anisotropic impure system is reported. It is found that both isotropic and anisotropic pure systems are unstable to the addition of random impurities for n nC identical to 4-2 epsilon +O( epsilon 2). Singularities in the epsilon expansions for the exponents of the random fixed points are shown to be associated with the onset of focal behaviour (the acquisition of complex eigenvalues). These, together with the slow convergence of the expansions leads one to doubt the reliability of extrapolations to three dimensions.