Type Of Critical Behavior Research Articles

Ising model with varying spin strength on a scale-free network: scaling functions and critical amplitude ratios

Recently, a novel model to describe ordering in systems comprising agents which, although matching in their binarity (i.e., maintaining the iconic Ising features of “+” or “–”, “up” or “down”, “yes” or “no”), still differing in their strength was suggested [Krasnytska et al., J. Phys. Complex., 2020, 1, 035008]. The model was analyzed for a particular case when agents are located on sites of a scale-free network and agent strength is a random variable governed by a power-law decaying distribution. For the annealed network, the exact solution shows a rich phase diagram with different types of critical behavior and new universality classes. This paper continues the above studies and addresses the analysis of scaling functions and universal critical amplitude ratios for the model on a scale-free network.

Unconventional criticality, scaling breakdown, and diverse universality classes in the Wilson-Cowan model of neural dynamics.

The Wilson-Cowan model constitutes a paradigmatic approach to understanding the collective dynamics of networks of excitatory and inhibitory units. It has been profusely used in the literature to analyze the possible phases of neural networks at a mean-field level, e.g., assuming large fully connected networks. Moreover, its stochastic counterpart allows one to study fluctuation-induced phenomena, such as avalanches. Here we revisit the stochastic Wilson-Cowan model paying special attention to the possible phase transitions between quiescent and active phases. We unveil eight possible types of such transitions, including continuous ones with scaling behavior belonging to known universality classes-such as directed percolation and tricritical directed percolation-as well as six distinct ones. In particular, we show that under some special circumstances, at a so-called "Hopf tricritical directed percolation" transition, rather unconventional behavior is observed, including the emergence of scaling breakdown. Other transitions are discontinuous and show different types of anomalies in scaling and/or exhibit mixed features of continuous and discontinuous transitions. These results broaden our knowledge of the possible types of critical behavior in networks of excitatory and inhibitory units and are, thus, of relevance to understanding avalanche dynamics in actual neuronal recordings. From a more general perspective, these results help extend the theory of nonequilibrium phase transitions into quiescent or absorbing states.

Dimensional transmutation and nonconventional scaling behavior in a model of self-organized criticality

This paper addresses two unusual scaling regimes (types of critical behavior) predicted by the field-theoretic renormalization group analysis for a self-organized critical system with turbulent motion of the environment. The system is modeled by the anisotropic stochastic equation for a “running sandpile” introduced by Hwa and Kardar in [Phys. Rev. Lett. 62, 1813 (1989)]. The turbulent motion is described by the isotropic Kazantsev–Kraichnan “rapid-change” velocity ensemble for an incompressible fluid. The original Hwa–Kardar equation allows for independent scaling of the spatial coordinates [Formula: see text] (the coordinate along the preferred dimension) and [Formula: see text] (the coordinates in the orthogonal subspace to the preferred direction) that becomes impossible once the isotropic velocity ensemble is coupled to the equation. However, it is found that one of the regimes of the system’s critical behavior (the one where the isotropic turbulent motion is irrelevant) recovers the anisotropic scaling through “dimensional transmutation.” The latter manifests as a dimensionless ratio acquiring nontrivial canonical dimension. The critical regime where both the velocity ensemble and the nonlinearity of the Hwa–Kardar equation are relevant simultaneously is also characterized by “atypical” scaling. While the ordinary scaling with fixed IR irrelevant parameters is impossible in this regime, the “restricted” scaling where the times, the coordinates, and the dimensionless ratio are scaled becomes possible. This result brings to mind scaling hypotheses modifications (Stell’s weak scaling or Fisher’s generalized scaling) for systems with significantly different characteristic scales.

Universal fluctuations and squeezing in a generalized Dicke model near the superradiant phase transition

In a view of recent proposals for the realization of anisotropic light-matter interaction in such platforms as (i) non-stationary or inductively and capacitively coupled superconducting qubits, (ii) atoms in crossed fields and (iii) semiconductor heterostructures with spin-orbital interaction, the concept of generalized Dicke model, where coupling strengths of rotating wave and counter-rotating wave terms are unequal, has attracted great interest. For this model, we study photon fluctuations in the critical region of normal-to-superradiant phase transition when both the temperatures and numbers of two-level systems are finite. In this case, the superradiant quantum phase transition is changed to a fluctuational region in the phase diagram that reveals two types of critical behaviors. These are regimes of Dicke model (with discrete $\mathbb{Z}_2$ symmetry), and that of (anti-) and Tavis-Cummings $U(1)$ models. We show that squeezing parameters of photon condensate in these regimes show distinct temperature scalings. Besides, relative fluctuations of photon number take universal values. We also find a temperature scales below which one approaches zero-temperature quantum phase transition where quantum fluctuations dominate. Our effective theory is provided by a non-Goldstone functional for condensate mode and by Majorana representation of Pauli operators. We also discuss Bethe ansatz solution for integrable $U(1)$ limits.

Dynamic critical behavior of the two-dimensional Ising model with nonextensive statistics.

The dynamic critical behavior of the two-dimensional Ising model with nonextensive Tsallis statistics has been studied. The values of the dynamic critical index z as well as the values of the indices ν and β for different values of the deformation parameter q have been obtained. The emergence of a new type of critical behavior has been revealed.

Universality Classes of the Hwa-Kardar Model with Turbulent Advection

Self-organized critical system in turbulent fluid environment is studied with the renormalization group analysis. The system is modelled by the anisotropic stochastic differential equation for a coarse-grained field proposed by Hwa and Kardar [Phys. Rev. Lett. 62, 1813 (1989)]. The turbulent motion of the environment is described by the anisotropic d-dimensional velocity ensemble based on the one introduced by Avellaneda and Majda [Commun. Math. Phys. 131, 381 (1990)] and modified to include dependence on time (finite correlation time). Renormalization group analysis reveals three universality classes (types of critical behavior) differentiated by the parameters of the system.

Specific heat and partition function zeros for the dimer model on the checkerboard B lattice: Finite-size effects

There are three possible classifications of the dimer weights on the bonds of the checkerboard lattice and they are denoted as checkerboard A, B, and C lattices [Phys. Rev. E 91, 062139 (2015)PLEEE81539-375510.1103/PhysRevE.91.062139]. The dimer model on the checkerboard B and C lattices has much richer critical behavior compared to the dimer model on the checkerboard A lattice. In this paper we study in full detail the dimer model on the checkerboard B lattice. The dimer model on the checkerboard B lattice has two types of critical behavior. In one limit this model is the anisotropic dimer model on rectangular lattice with algebraic decay of correlators and in another limit it is the anisotropic generalized Kasteleyn model with radically different critical behavior. We analyze the partition function of the dimer model on a 2M×2N checkerboard B lattice wrapped on a torus. We find very unusual behavior of the partition function zeros and the specific heat of the dimer model. Remarkably, the partition function zeros of finite-size systems can have very interesting structures, made of rings, concentric circles, radial line segments, or even arabesque structures. We find out that the number of the specific heat peaks and the number of circles of the partition function zeros increases with the system size. The lattice anisotropy of the model has strong effects on the behavior of the specific heat, dominating the relation between the correlation length exponent ν and the shift exponent λ, and λ is generally unequal to 1/ν (λ≠1/ν).

Oscillon in Einstein-scalar system with double well potential and its properties.

The dynamical evolution of self-interacting scalar field has many nontrivial behaviors, which tell us many lessons in a nonlinear dynamics. On Minkowski spacetime, the scalar field with double well potential has localized, non-singular, time-dependent, long-lived solutions, which are called oscillons. The lifetime of the oscillon depends on the initial conditions. Furthermore, when the initial parameter is fine-tuned, oscillons can be infinitely, and type I critical behavior is observed. Here, we investigate the Einstein-scalar system with double well potential. We show that oscillons exist in this system, and discuss the behavior when the initial parameter is fine-tuned. Our results suggests that a new type of critical behavior appears in this theory.

Self-gravitating oscillons and new critical behavior

The dynamical evolution of self-interacting scalars is of paramount importance in cosmological settings, and can teach us about the content of Einstein's equations. In flat space, nonlinear scalar field theories can give rise to localized, non-singular, time-dependent, long-lived solutions called {\it oscillons}. Here, we discuss the effects of gravity on the properties and formation of these structures, described by a scalar field with a double well potential. We show that oscillons continue to exist even when gravity is turned on, and we conjecture that there exists a sequence of critical solutions with infinite lifetime. Our results suggest that a new type of critical behavior appears in this theory, characterized by modulations of the lifetime of the oscillon around the scaling law and the modulations of the amplitude of the critical solutions.

Two-dimensional disordered Ising model within nonextensive statistics

In this work, the two-dimensional disordered Ising model with nonextensive Tsallis statistics has been studied for the first time. The critical temperatures and critical indices have been determined for both disordered and uniform models. A new type of critical behavior has been revealed for the disordered model with nonextensive statistics. It has been shown that, within the nonextensive statistics of the two-dimensional Ising model, the Harris criterion is also valid.

Different types of critical behavior in conservatively coupled Hénon maps.

We study the dynamics of two conservatively coupled Hénon maps at different levels of dissipation. It is shown that the decrease of dissipation leads to changes in the structure of the parameter plane and the scenarios of transition to chaos compared to the case of infinitely strong dissipation. Particularly, the Feigenbaum line becomes divided into several fragments. Some of these fragments have critical points of different types, namely, of C and H type, as their terminal points. Also the mechanisms of formation of these Feigenbaum line ruptures are described.

Damage spreading method in studies of the critical behavior of disordered systems

The computer simulations of the critical dynamics of the structurally disordered three-dimensional Ising model are performed by the damage spreading method. For the systems with spin densities p = 0.6 and 0.8, we calculate the critical temperature and dynamic critical exponent z characterizing the behavior of relaxation properties near the critical point. The analysis of the results demonstrates the nonuniversality of the critical dynamics in the disordered Ising model. To interpret such dynamics, we introduce the concept of two universal types of critical behavior corresponding to weak and strong disorder, respectively.

Effects of turbulent mixing on critical behaviour: renormalization-group analysis of the Potts model

The critical behaviour of a system, subjected to strongly anisotropic turbulent mixing, is studied by means of the field-theoretic renormalization group. Specifically, the relaxational stochastic dynamics of a non-conserved multicomponent order parameter of the Ashkin–Teller–Potts model, coupled to a random velocity field with prescribed statistics, is considered. The velocity is taken to be Gaussian, white in time, with a correlation function of the form ∝δ(t − t′)/|k⊥|d − 1 + ξ, where k⊥ is the component of the wave vector, perpendicular to the distinguished direction (‘direction of the flow’)—the d-dimensional generalization of the ensemble was introduced by Avellaneda and Majda (1990 Commun. Math. Phys. 131 381) within the context of passive scalar advection. This model can describe a rich class of physical situations. It is shown that, depending on the values of the parameters that define the self-interaction of the order parameter and the relation between the exponent ξ and the space dimension d, the system exhibits various types of large-scale scaling behaviour, associated with different infrared attractive fixed points of the renormalization-group equations. In addition to known asymptotic regimes (critical dynamics of the Potts model and passively advected field without self-interaction), the existence of a new, non-equilibrium and strongly anisotropic, type of critical behaviour (universality class) is established, and the corresponding critical dimensions are calculated to the leading order of the double expansion in ξ and ε = 6 − d (one-loop approximation). The scaling appears to be strongly anisotropic in the sense that the critical dimensions related to the directions parallel and perpendicular to the flow are essentially different.

Effects of turbulent transfer on critical behavior

Using the field theory renormalization group, we study the critical behavior of two systems subjected to turbulent mixing. The first system, described by the equilibrium model A, corresponds to the relaxational dynamics of a nonconserved order parameter. The second system is the strongly nonequilibrium reaction-diffusion system, known as the Gribov process or directed percolation process. The turbulent mixing is modeled by the stochastic Navier-Stokes equation with a random stirring force with the correlator ∞ δ(t − t′)p 4−d−y, where p is the wave number, d is the space dimension, and y is an arbitrary exponent. We show that the systems exhibit various types of critical behavior depending on the relation between y and d. In addition to known regimes (original systems without mixing and a passively advected scalar field), we establish the existence of new strongly nonequilibrium universality classes and calculate the corresponding critical dimensions to the first order of the double expansion in y and ɛ = 4 − d (one-loop approximation).

Effects of turbulent mixing on critical behaviour in the presence of compressibility: renormalization group analysis of two models

Critical behaviour of two systems, subjected to the turbulent mixing, is studied by means of the field theoretic renormalization group. The first system, described by the equilibrium model A, corresponds to the relaxational dynamics of a non-conserved order parameter. The second one is the strongly non-equilibrium reaction–diffusion system known as Gribov process and equivalent to the Reggeon field theory. The turbulent mixing is modelled by the Kazantsev–Kraichnan ‘rapid-change’ ensemble: time-decorrelated Gaussian velocity field with the power-like spectrum ∝k−d − ξ. Effects of compressibility of the fluid are studied. It is shown that, depending on the relation between the exponent ξ and the spatial dimension d, both the systems exhibit four different types of critical behaviour, associated with four possible fixed points of the renormalization group equations. Three fixed points correspond to known regimes: Gaussian fixed point, original model without mixing and passively advected scalar field. The most interesting fourth point corresponds to a new type of critical behaviour, in which both nonlinearity and turbulent mixing are relevant, and the critical exponents depend on d, ξ and the degree of compressibility. The critical exponents and regions of stability for all the regimes are calculated in the leading order of the double expansion in two parameters ξ and ε = 4 − d. For both models, compressibility enhances the role of the nonlinear terms in the dynamical equations: the region in the ε–ξ plane, where the new nontrivial regime is stable, is getting much wider as the degree of compressibility increases. For the incompressible fluid, the most realistic values d = 3 and ξ = 4/3 (Kolmogorov turbulence) lie in the region of stability of the passive scalar regime. If the compressibility becomes strong enough, the crossover in the critical behaviour occurs, and these values of d and ξ fall into the region of stability of the new regime, where both advection and the nonlinearity are important. In its turn, turbulent transfer becomes more efficient due to combined effects of the mixing and the nonlinear terms.

Boundary magnetization of the generalized McCoy-Wu type random Ising models

This paper investigates the generalized McCoy-Wu type random Ising model. It is found that boundary magnetization behaves as (Tc − T)β', β' = 1 just below the critical temperature. From numerical analyses it is revealed that this type of critical behavior, β' = 1, is universal for any type of randomness. This universal behavior is explained in terms of the destruction of the devil's staircase structure.

Square lattice hard-core bosons within the random phase approximation

Using random phase approximation we build finite-temperature phase diagrams and calculate thermodynamic functions of a square lattice hard-core boson model with nearest (nn) and next-nearest-neighbor (nnn) interactions in terms of equivalent to it an anisotropic spin-1/2 $XXZ$ model in a magnetic field. The system undergoes liquid-solid phase transitions that can be either of the first or second order. Depending on the hopping value and ratio between nn and nnn interactions the system displays two types of critical behavior. The line of the first-order transitions terminates in a bicritical end point inside the solid phase or ends in a tricritical point continuously giving way to the second-order phase transition line. The connection of the hopping and the ratio between the nn and nnn interactions with criticality of the system is investigated. The critical line separating the regions of specific critical behavior is built. Reentrant liquid-solid-liquid phase transitions are found and discussed.

A vortex solid-to-liquid transition with fully anisotropic scaling

The vortex solid-to-liquid transition has been studied in heavy ion irradiated untwinned single crystals of YBa2Cu3O7-δ with an inclined applied magnetic field. For magnetic fields tilted at angles about 45° away from the columnar defects, we find that the electric resistivity in the vortex liquid regime approaches zero with power laws in the reduced temperature T — Tc that have different exponents in all three spatial directions. Since the symmetry in the problem has been broken in two non-collinear directions by i) the direction of the columnar defects and ii) the direction of the applied magnetic field, our findings give evidence for a new type of critical behavior with fully anisotropic critical exponents. A possible view of the vortex topology for the transition is also suggested.

Cluster geometry and survival probability in systems driven by reaction–diffusion dynamics

We consider a reaction–diffusion model incorporating the reactions A→ϕ, A→2A and 2A→3A. Depending on the relative rates for sexual and asexual reproduction of the quantity A, the model exhibits either a continuous or first-order absorbing phase transition to an extinct state. A tricritical point separates the two phase lines. While we comment on this critical behaviour, the main focus of the paper is on the geometry of the population clusters that form. We observe the different cluster structures that arise at criticality for the three different types of critical behaviour and show that there exists a linear relationship for the survival probability against initial cluster size at the tricritical point only.

DIRTY BOSONS: TWENTY YEARS LATER

A concise and somewhat personal review of the problem of superfluidity and quantum criticality in regular and disordered interacting Bose systems is given, concentrating on general features and important symmetries that are exhibited in different parts of the phase diagram, and that govern the different possible types of critical behavior. A number of exact results for various insulating phase boundaries, which may be used to constrain the results of numerical simulations, can be derived using large rare region type arguments. The nature of the insulator-superfluid transition is explored through general scaling arguments, exact model calculations in one dimension, numerical results in two dimensions, and approximate renormalization group results in higher dimensions. Experiments on 4 He adsorbed in porous Vycor glass, on thin film superconductors, and magnetically trapped atomic vapors in a periodic optical potential are used to illustrate many of the concepts.