Universality Class Of Critical Behavior Research Articles

Фазовые переходы в фрустрированных моделях Изинга (О б з о р)

The current state of research on phase transitions and critical phenomena in frustration is considered. modeled Ising models. The results obtained on the basis of highly efficient algorithms are discussed. Monte Carlo method for Ising models on lattices of various dimensions and types. A main attention given to the results of studying phase transitions, critical and thermodynamic properties of models Ising, taking into account the exchange interactions of the first and second neighbors, as well as the external magnetic fields. Phase diagrams of the dependence of the critical temperature on the value of the exchange interactions of second neighbors and an external magnetic field. The regions of observation of phase transitions of the first and second kind. The universality classes of critical behavior are established. For some models have ranges with non-universal critical behavior. Discussed features of the critical behavior of frustrated Ising models.

Criticality in Spreading Processes without Timescale Separation and the Critical Brain Hypothesis

Spreading processes on networks are ubiquitous in both human-made and natural systems. Understanding their behavior is of broad interest: from the control of epidemics to understanding brain dynamics. While in some cases there exists a clear separation of timescales between the propagation of a single spreading cascade and the initiation of the next—such that spreading can be modeled as directed percolation or a branching process—there are also processes for which this is not the case, such as spiking cascades in neural networks. For a large class of relevant network topologies, we show here that in such a scenario the nature of the overall spreading fundamentally changes. This change manifests itself in a transition between different universality classes of critical behavior, which determines the onset and the properties of neural activity turning epileptic, for example. We present analytical results in the mean-field limit, giving the critical line along which scale-free behavior can be observed. The two limits of this critical line correspond to the universality classes of directed and undirected percolation, respectively. Outside these two limits, this duality manifests itself in the appearance of critical exponents from the universality classes of both directed and undirected percolation. We find that the transition between these exponents is governed by a competition between merging and propagation of activity, and we identify an appropriate scaling relationship for the transition point. Finally, we show that commonly used measures, such as the branching ratio and dynamic susceptibility, fail to establish criticality in the absence of timescale separation, calling for a reanalysis of criticality in the brain.10 MoreReceived 5 June 2020Revised 22 February 2021Accepted 12 April 2021Corrected 25 June 2021DOI:https://doi.org/10.1103/PhysRevX.11.021059Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasNetwork phase transitionsNeuronal avalanchesNonequilibrium statistical mechanicsSpreadingNetworksStatistical PhysicsBiological Physics

Integrable quantum spin chains with free fermionic and parafermionic spectrum

We present a general study of the large family of exact integrable quantum chains with multispin interactions introduced recently in \cite{AP2020}. The exact integrability follows from the algebraic properties of the energy density operators defining the quantum chains. The Hamiltonians are characterized by a parameter $p=1,2,\dots$ related to the number of interacting spins in the multispin interaction. In the general case the quantum spins are of infinite dimension. In special cases, characterized by the parameter $N=2,3,\ldots$, the quantum chains describe the dynamics of $Z(N)$ quantum spin chains. The simplest case $p=1$ corresponds to the free fermionic quantum Ising chain ($N=2$) or the $Z(N)$ free parafermionic quantum chain. The eigenenergies of the quantum chains are given in terms of the roots of special polynomials, and for general values of $p$ the quantum chains are characterized by a free fermionic ($N=2$) or free parafermionic ($N>2$) eigenspectrum. The models have a special critical point when all coupling constants are equal. At this point the ground-state energy is exactly calculated in the bulk limit, and our analytical and numerical analyses indicate that the models belong to universality classes of critical behavior with dynamical critical exponent $z = (p+1)/N$ and specific-heat exponent $\alpha = \max\{0,1-(p+1)/N\}$.

Influence of initial states and structure defects on non-equilibrium critical behavior of 3D and 2D Ising models

Results of Monte Carlo description of features of non-equilibrium critical behavior in three- and two-dimensional Ising models conditioned by presence of structural quenched disorder are presented. It was revealed that the increase in the concentration of defects is accompanied by the strengthening of the aging effects manifested as the slowing down of correlation and relaxation processes in structurally disordered systems as compared to the pure system. Non-equilibrium initial states begin to increasingly more strongly influence peculiar features and characteristics of the system’s evolution. It was shown that limiting fluctuation-dissipation ratio values X∞ characterizing the degree of system departure from equilibrium satisfy inequality X∞ < 1 and depend on the universality class of non-equilibrium critical behavior to which they belong: one of these classes corresponds to the high-temperature, and the other to the low-temperature initial state of the system. Values of these non-equilibrium characteristics in 3D Ising model demonstrate belonging to universality classes of critical behavior of weakly and strongly diluted systems, but non-equilibrium characteristics in 2D Ising model depend continuously on concentration of defects owing to the crossover effects in the percolation behavior.

Критическое поведение систем с конкуренцией между близкодействием и дальнодействием

Critical behaviour of a range of ferromagnetic materials deviates from the predictions of the Ising, XY and Heisenberg models. Additional long-range forces competing with regular exchange interaction may explain this deviation. These competing interactions lead to new universality classes of critical behaviour. The paper uses the field theory approach to investigate critical behaviour in those systems in which long-range and short-range forces compete. We consider the case when a power function of distance r-D-σ, when 1.5 &lt; σ &lt; 2.0, can describe the long-range forces. There exists a distinctive critical behaviour mode for these values. We derived vertex functions using a two-loop approximation directly in three-dimensional space (D = 3) and, for all values, obtained a linear approximation of asymptotic series in terms of long-range interaction parameters. We applied the Pade --- Borel summation technique to these asymptotic series. We computed stable fixed points and critical exponents as functions of long-range interaction parameters for low relativeefficiency of the long-range interaction. We investigated how critical exponents depend on the factor in the power law and relative long-range interaction intensity. We compared our results to the critical exponent values found experimentally for manganites. We used the experimental critical exponent γ values to compute long-range interaction parameters and then used the long-range interaction parameters to derive the ß exponent values, which we then compared to the experimental values. We show good agreement between our theoretical results and experimental data.

The $$\epsilon $$ ϵ Expansion and Universality in Three Dimensions

It has been observed that the clasification into universality classes of critical behaviour, as established by perturbative renormalization group in the viscinity of four or six dimensions of space by the epsilon expansion, remains valid down to three dimensions in all known cases, even when purturbative renormalisation group fails in three dimensions. In this paper we argue that this classification into universality classes remains true in lower dimensions of space, even when purturbative renormalisation group fails, because of the well known phenomenon of eigenvalue repulsion.

Phase transitions and critical properties in the antiferromagnetic Ising model on a layered triangular lattice with allowance for intralayer next-nearest-neighbor interactions

The phase transitions (PTs) and critical properties of the antiferromagnetic Ising model on a layered (stacked) triangular lattice have been studied by the Monte Carlo method using a replica algorithm with allowance for the next-nearest-neighbor interactions. The character of PTs is analyzed using the histogram technique and the method of Binder cumulants. It is established that the transition from the disordered to paramagnetic phase in the adopted model is a second-order PT. Static critical exponents of the heat capacity (α), susceptibility (γ), order parameter (β), and correlation radius (ν) and the Fischer exponent η are calculated using the finite-size scaling theory. It is shown that (i) the antiferromagnetic Ising model on a layered triangular lattice belongs to the XY universality class of critical behavior and (ii) allowance for the intralayer interactions of next-nearest neighbors in the adopted model leads to a change in the universality class of critical behavior.

The inactive–active phase transition in the noisy additive (exclusive-or) probabilistic cellular automaton

We investigate the inactive–active phase transition in an array of additive (exclusive-or) cellular automata (CA) under noise. The model is closely related with the Domany-Kinzel (DK) probabilistic cellular automaton (PCA), for which there are rigorous as well as numerical estimates on the transition probabilities. Here, we characterize the critical behavior of the noisy additive cellular automaton by mean field analysis and finite-size scaling and show that its phase transition belongs to the directed percolation universality class of critical behavior. As a by-product of our analysis, we argue that the critical behavior of the noisy elementary CA 90 and 102 (in Wolfram’s enumeration scheme) must be the same. We also perform an empirical investigation of the mean field equations to assess their quality and find that away from the critical point (but not necessarily very far away) the mean field approximations provide a reasonably good description of the dynamics of the PCA.

Phase transitions and critical characteristics in the layered antiferromagnetic Ising model with next-nearest-neighbor intralayer interactions

Phase transitions and critical characteristics of the layered antiferromagnetic Ising model in the case of a cubic lattice with next-nearest-neighbor intralayer interactions are studied in the framework of the Monte Carlo method implementing the replica algorithm. The characteristics of the phase transitions are analyzed within the histogram method and with the Binder cumulants. For the model under study, it is found that the transition from the superantiferromagnetic phase to the paramagnetic one is a second order phase transition. The static critical exponents for the specific heat α, susceptibility γ, order parameter β, correlation radius ν, and the Fisher exponent η are calculated using the finite-size scaling theory. It is shown that the three-dimensional Ising model for the cubic lattice with next-nearest-neighbor interactions belongs to the same universality class of critical behavior as the completely frustrated three-dimensional Ising model.

Mesoscale phenomena in solutions of 3-methylpyridine, heavy water, and an antagonistic salt

We have investigated controversial issues regarding the mesoscale behavior of 3-methylpyridine (3MP), heavy water, and sodium tetraphenylborate (NaBPh4) solutions by combining results obtained from dynamic light scattering (DLS) and small-angle neutron scattering (SANS). We have addressed three questions: (i) what is the origin of the mesoscale inhomogeneities (order of 100 nm in size) manifested by the "slow mode" in DLS? (ii) Is the periodic structure observed from SANS an inherent property of this system? (iii) What is the universality class of critical behavior in this system? Our results confirm that the "slow mode" observed from DLS experiments corresponds to long-lived, highly stable mesoscale droplets (order of 100 nm in size), which occur only when the solute (3MP) is contaminated by hydrophobic impurities. SANS data confirm the presence of a periodic structure with a periodicity of about 10 nm. This periodic structure cannot be eliminated by nanopore filtration and thus is indeed an inherent solution property. The critical behavior of this system, in the range of concentration and temperatures investigated by DLS experiments, indicates that the criticality belongs to the universality class of the 3-dimensional Ising model.

Entanglement entropy of the low-lying excited states and critical properties of an exactly solvable two-leg spin ladder with three-spin interactions

In this work, we investigate an exactly solvable two-leg spin ladder with three-spin interactions. We obtain analytically the finite-size corrections of the low-lying energies and determine the central charge as well as the scaling dimensions. The model considered in this work is in the same universality class of critical behavior of the $XX$ chain with central charge $c=1$. By using the correlation matrix method, we also study the finite-size corrections of the R\'enyi entropy of the ground state and of the excited states. Our results are in agreement with the predictions of the conformal field theory.

Exact field-driven interface dynamics in the two-dimensional stochastic Ising model with helicoidal boundary conditions

We investigate the interface dynamics of the two-dimensional stochastic Ising model in an external field under helicoidal boundary conditions. At sufficiently low temperatures and fields, the dynamics of the interface is described by an exactly solvable high-spin asymmetric quantum Hamiltonian that is the infinitesimal generator of the zero range process. Generally, the critical dynamics of the interface fluctuations is in the Kardar–Parisi–Zhang universality class of critical behavior. We remark that a whole family of RSOS interface models similar to the Ising interface model investigated here can be described by exactly solvable restricted high-spin quantum XXZ-type Hamiltonians.

Monte Carlo investigation of the critical behavior of Stavskaya’s probabilistic cellular automaton

Stavskaya's model is a one-dimensional probabilistic cellular automaton (PCA) introduced in the end of the 1960s as an example of a model displaying a nonequilibrium phase transition. Although its absorbing state phase transition is well understood nowadays, the model never received a full numerical treatment to investigate its critical behavior. In this Brief Report we characterize the critical behavior of Stavskaya's PCA by means of Monte Carlo simulations and finite-size scaling analysis. The critical exponents of the model are calculated and indicate that its phase transition belongs to the directed percolation universality class of critical behavior, as would be expected on the basis of the directed percolation conjecture. We also explicitly establish the relationship of the model with the Domany-Kinzel PCA on its directed site percolation line, a connection that seems to have gone unnoticed in the literature so far.

Influence of Ag doping on the critical behavior of the heat capacity of manganites

The critical behavior of the heat capacity of Ag doped manganites is studied. Changing regularities of universal critical parameters near the critical point are determined. All investigated samples up to t min ≈ 10−3 correspond to 3D Heisenberg universality class of the critical behavior. The universality class of critical behavior of La1 – x Ag y MnO3 (y ≤ x) heat capacity is shown to be Ag-concentration independent.

Monte Carlo investigation of the critical properties of a three-dimensional frustrated Heisenberg model on a triangular lattice

The Monte Carlo replica method is used to investigate the critical properties of a three-dimensional frustrated antiferromagnetic Heisenberg model on a triangular lattice. The static magnetic and chiral critical exponents are calculated within the theory of finite-dimensional scaling: specific heat α=0.05(2); magnetization β=0.30(1), βk=0.52(2); susceptibility γ=1.36(2), γk=0.93(3); and, correlation radius ν=0.64(1), νk=0.64(2). The critical Fisher indices η=−0.06(3) and ηk=0.63(4) for this model are calculated for the first time. It is shown that the three-dimensional frustrated Heisenberg model on a triangular lattice forms a new universality class of critical behavior. It is found that the universality class of the antiferromagnetic Heisenberg model on a triangular lattice depends on the type of interlayer exchange interaction.

Critical behavior of the specific heat of manganites La1−xAgxMnO3 (x=0.1,0.15,0.2) near the Curie point

An investigation of the critical behavior of the specific heat of the manganites La1−xAgxMnO3 ×(x=0.1,0.15,0.2) near the Curie temperature is carried out. The behavior of the universal critical parameters near the phase transition point is established. All of the samples studied correspond to the ferromagnetic Heisenberg 3D universality class of critical behavior, with the critical exponent α=−0.115, −0.106, and −0.106 for La0.9Ag0.1MnO3, La0.85Ag0.15MnO3, and La0.8Ag0.2MnO3, respectively. It is shown that the universality class of the critical behavior of the specific heat of the manganites La1−xAgxMnO3 is independent of the silver concentration.

Investigation of the effect of frustration on the critical properties of the 3D Heisenberg antiferromagnetic model

The critical properties of the 3D frustrated Heisenberg model on a layered triangular lattice with a variable interlayer exchange interaction are investigated for the ratio of the constants of the interlayer (J′) and intralayer ( J ) interactions that ranges within the interval R = | J′/J| = 0.01…1. The finite-dimensional scaling theory is applied for the calculation of all of the main static magnetic and chiral critical indexes. It is shown that the 3D frustrated Heisenberg model on a layered triangular lattice forms a new universality class of critical behavior. It is found that the universality class of this model’s critical behavior is retained for values of R as small as R = 0.05.

Static critical behavior of 3D frustrated Heisenberg model on stacked triangular lattice with variable interlayer exchange coupling

Several Monte Carlo algorithms are used to examine the critical behavior of the 3D frustrated Heisenberg model on stacked triangular lattice with variable interlayer exchange coupling for values of the interlayer-to-intralayer exchange ratio R = |′/J| in the interval between 0.01 and 1.0. A finite-size scaling technique is used to calculate the static magnetic and chiral critical exponents α (specific heat), γ and γk (susceptibility), β and βk(magnetization), ν and νk(correlation length), and the Fisher exponent η. It is shown that 3D frustrated Heisenberg models on stacked triangular lattice with R > 0.05 constitute a new universality class of critical behavior. At lower R, a crossover from 3D to 2D critical behavior is observed.

Critical properties of small magnetic particles of YFeO3

Different ways of modeling small magnetic particles of yttrium orthoferrite are proposed. Monte Carlo methods are used to study the static critical properties of these models. The conventional formalism of power-law functions is used to calculate the main static critical exponents for the proposed models, and their universality classes are determined. It is shown that models of small magnetic particles of YFeO3 are described mainly by the Ising universality class of critical behavior. The results can explain the contradiction between the experimentally observed and theoretically predicted pictures of critical behavior of small magnetic particles of yttrium orthoferrite.

Criticality on networks with topology-dependent interactions

Weighted scale-free networks with topology-dependent interactions are studied. It is shown that the possible universality classes of critical behavior, which are known to depend on topology, can also be explored by tuning the form of the interactions at fixed topology. For a model of opinion formation, simple mean field and scaling arguments show that a mapping gamma'=(gamma-mu)(1-mu) describes how a shift of the standard exponent gamma of the degree distribution can absorb the effect of degree-dependent pair interactions J(ij) proportional to (k(i)k(j))(-mu), where k(i) stands for the degree of vertex i. This prediction is verified by extensive numerical investigations using the cavity method and Monte Carlo simulations. The critical temperature of the model is obtained through the Bethe-Peierls approximation and with the replica technique. The mapping can be extended to nonequilibrium models such as those describing the spreading of a disease on a network.