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An approach developed in the author’s recent paper [V.N. Bondarev, Phys. Rev. E 77, R050103 (2008)], where the independent calculation of three critical exponents for a fluid was demonstrated, is generalized to the case of the critical isotherm. This has allowed us to calculate, independently, three more critical exponents: δ = 4, μ = 5/12, and α = 1/8. These “non-classical” values are in good agreement with those determined experimentally and are rather close to the 3-dimensional Ising ones. In addition, on the basis of this approach a solution of the so-called Yang-Yang problem concerning the “pressure” and the “chemical potential” contributions to the critical singularity of the isochoric heat capacity of a fluid is proposed. Also, the so-called “corrections to scaling” terms are considered in the framework of the developed theory and the corresponding exponents at the critical isochore and the critical isotherm are calculated. The results obtained provide a natural perspective for including the non-classical (of the Ising-type) description of fluid critical properties into the conventional theory of liquids.

We have investigated the critical behavior of Fe2.85GeTe2 single crystals near the paramagnetic-ferromagnetic transition by bulk dc magnetization measurements. The critical exponents β, γ and δ, obtained from modified Arrott plot, Kouvel-Fisher method, and critical isothermal magnetization analysis, could fulfill the Widom scaling law. The self-consistency and reliability of these exponents are further verified by the magnetic state equations below and above the Curie temperature at high magnetic field. In addition, the exchange distance deduced from the susceptibility exponent is shown to decay as [Formula: see text]. Based on the observations, we suggest that the competition between direct magnetic exchange and Coulombic and/or RKKY interactions should be responsible for the intrinsic magnetism in this system.

In this work, the universal curve, critical behavior and magnetocaloric effect of the intermetallic compound Mn0.92Sn0.08CoGe around room temperature have been investigated. Magnetic data indicate that Mn0.92Sn0.08CoGe exhibits the characteristic of second-order transition around the room temperature 286 K. Moreover, the universal curve, well fitted by a Lorentz function, further proves second-order nature of the transition. The critical exponents obtained from various techniques such as modified Arrott plot (MAP), Kouvel-Fisher (KF) method, and critical isotherm analysis (M(T C )) are found to basically fulfill the Widom scaling relation: δ = 1 + γ/β and obey the scaling equation of M (H, e) = e β f ±(H/e β+γ ), revealing the extracted critical exponents are reliable. The obtained critical exponents are close to the prediction of the mean-field model, suggesting the existence of long-range magnetic interactions coupling with the exchange distance decaying as J (r) ≈ r −4.4.

A long-standing issue in a three-dimensional Ising spin glass (3DISG) is to settle the critical exponents. Numerical simulations and a conventional finite-size-scaling analysis have given controversial estimates, in particular, for the value of thermal exponent ν. Here a recently proposed extended scaling scheme is reviewed and illustrated for the canonical two-dimensional Ising ferromagnet. By applying the extended scaling to our extensive Monte Carlo data for 3DISG, consistent estimates of critical parameters are obtained in contrast to previous analyses.

We present a review of the recent progress on percolation scaling limits in two dimensions. In particular, we consider the convergence of critical crossing probabilities to Cardy's formula and of the critical exploration path to chordal stochastic Loewner evolution (SLE6), the full scaling limit of critical cluster boundaries, and near-critical scaling limits.

Ultrasonic attenuation spectra between 20 kHz and 3 GHz of the nitrobenzene–isooctane mixture of critical composition have been analyzed to show that they contain noncritical relaxation terms in addition to the critical term. The parameter values of the noncritical contributions obtained thereby are used in a re-evaluation of smallband attenuation data from the literature. These data, measured at a large number of temperatures near the critical, are most suitable for the determination of the scaling function in the critical dynamics. The procedure allows to verify the empirical scaling function of the Bhattacharjee–Ferrell dynamic scaling theory without an adjustable parameter.

In this work we study the critical behavior of a model that simulates the propagation of an epidemic process over a population. We simulate the model on two-dimensional finite lattices to determine the critical density of the diffusive population. A finite size scaling analysis is employed to determine the order parameter and correlation length critical exponents.

Purpose – The paper presents an application of the scaling theory in a description of energy losses in soft magnetic materials in order to improve an agreement between measurements and theoretical models.Design/methodology/approach – The scaling theory allows the description of energy losses by a generalized homogenous function, which depends on scaling exponents α, β and amplitudes Γ(n). The values of the scaling exponents and amplitudes were estimated on the basis of measurement data of total energy losses.Findings – The main findings of the paper are: the linear relationships between the scaling exponents α and β, the data collapse of energy losses and the scaling laws for asymptotic exponents of energy losses derivatives.Research limitations/implications – The origin of the data collapse and the relationship between the scaling exponents will be the subject of further research with the aid of renormalization group methodPractical implications – The paper could be useful both for device designers and r...

Progress on the determination of the order of the fragmentation phase transition, the location of its critical point in the nuclear matter phase diagram, the values of the critical exponents that determine the universality class of the transition, and finite size scaling effects is discussed. Evidence for the presence of Zipf-Mandelbrot-scaling in the relative size of the largest clusters is examined, and the connection to the value of the critical exponent τ is established.

We report a study of the critical phenomena of perovskite-manganite compound La0.6Pr0.1Sr0.3MnO3 around the Curie temperature. Experimental results based on magnetic measurements using Banerjee criterion reveals that the sample exhibits a second-order paramagnetic–ferromagnetic transition. The critical behavior analysis and the Kouvel–Fisher method suggests that the critical phenomena around the critical point can be correctly described by the 3D-Heisenberg model. Critical exponents were estimated and found β¼0.35470.009 and γ¼1.26470.035 at TC¼325.570.443 K. The critical exponent δ is determined separately from the isothermal magnetization at TC and evaluated to δ¼4.93470.0004. These critical exponents obey the Widom scaling relation δ¼1þγ/β. Based on the critical exponents, the magnetization–field–temperature (M–H–T) data around TC collapses into two curves obeying the single scaling equation MH ; e ðÞ ¼ e jj β f 7 ðH= e jj β þ γ

We report the critical behavior in La0.5Sr0.5MnO3 perovskite oxide manganite using methods including modified Arrott plot, Kouvel–Fisher plot, and critical isotherm analysis. By means of the above techniques around the ferromagnetic–paramagnetic (FM–PM) transition, the magnetic data analyzed shows the sample attending the second-order magnetic phase transition with the critical parameters β = 0.553±0.020 with TC=352.02±0.63 K (from the spontaneous magnetization MS(T) below TC) and γ =1.169±0.058 with TC=352.49±0.85 K (from the inverse initial susceptibility χ 0 − 1 ( T ) above TC). The exponent δ =3,216 independently obtained from the critical magnetization isotherm satisfies the Widom scaling relation δ = 1 + β / γ . Moreover, the critical exponents are in the theoretically expected values for mean-field interaction model rather than the other universal theory and also follow the single scaling equation of M ( H , e ) e − β = f ± ( H / e β + γ ) . Furthermore, to estimate the spontaneous magnetization of La0.5Sr0.5MnO3 perovskite, we used the magnetic entropy change (−ΔSM), obtained from isothermal magnetization. The results of this approach are compared to results obtained from classical analysis using Arrott curves.

The critical properties of the mixed manganite La 0.67− x Y x Ba 0.23 Ca 0.1 MnO 3 with x =0.10 and x =0.15 around the paramagnetic (PM)-ferromagnetic (FM) phase transition were investigated through various techniques. These involved modified Arrott plots, Kouvel-Fisher method and Widom scaling relation. Magnetic data, analyzed in the critical region, using the above methods, yielded the critical exponents for ( x =0.10) La 0.57 Y 0.10 Ba 0.23 Ca 0.1 MnO 3 (β=0.312±0.002 and γ=1.147±0.003 at T C =299.23±0.05 K). Moreover, the estimated critical exponents of ( x =0.15) La 0.52 Y 0.15 Ba 0.23 Ca 0.1 MnO 3 were β=0.286±0.004 and γ=0.943±0.002 at T C =289.53±0.06 K. The critical exponents' values were close to the theoretical values of 3D-Ising model and tricritical mean-field model. These results suggested that the present composition should be close to a tricritical point in the La 0.67− x Y x Ba 0.23 Ca 0.1 MnO 3 phase diagram. Expressing the field dependence as Δ S M ∝ H n allowed us to establish a relationship between the exponent n and the critical exponents of the material and to propose a phenomenological universal curve for the field dependence of Δ S M .

La0.67–xDyxPb0.33MnO3 (x=0.00 and x=0.10) were elaborated by the solid state method and checked by X-ray diffraction. Close to magnetic temperature transition, the order transition and the critical behavior were investigated by dc magnetization measurements versus x composition. The critical properties were investigated through various techniques such as modified Arrott plot (MAP), Kouvel-Fisher (KF) method and critical isotherm (CI) analysis based on the data of static magnetic measurements recorded around the Curie temperature TC. The values of critical exponents (β and γ) estimated were found to lie between those predicted for a 3D-Ising model for x=0.00 and those of 3D-Heisenberg model for x=0.10. The reliability of the critical exponent's values was confirmed by the Widom scaling relation and the universal scaling hypothesis. The change in the universality class should be due to the increase of the Dy content.

The critical behavior of the single-crystal helimagnet MnSi is investigated by means of bulk dc magnetization at the boundary between the conical state and paramagnetic phase. We obtain the critical exponents $(\ensuremath{\beta}=0.242\ifmmode\pm\else\textpm\fi{}0.006,\ensuremath{\gamma}=0.915\ifmmode\pm\else\textpm\fi{}0.003$, and $\ensuremath{\delta}=4.734\ifmmode\pm\else\textpm\fi{}0.006)$, where the self-consistency and reliability are verified by the Widom scaling law and the scaling equation. The critical exponents of MnSi belong to the universality class of tricritical mean-field theory, which unambiguously indicates a tricritical phenomenon at the boundary between the first-order phase transition and the second-order one induced by the external magnetic field. The tricritical point (TCP) is determined as ${H}_{\mathrm{TCP}}\ensuremath{\approx}3200$ Oe at the critical temperature, consistent with the previous report [A. Bauer, M. Garst, and C. Pfleiderer, Phys. Rev. Lett. 110, 177207 (2013)]. The critical behavior suggests a long-range magnetic coupling with the exchange distance decaying as $J(r)\ensuremath{\approx}{r}^{\ensuremath{-}4.3}$ in MnSi.

We present a scaling theory for the entanglement spectrum under an external driving. Based on the static scaling of the Schmidt gap and the theory of finite-time scaling, we show that the Schmidt gap can signal the critical point and be used to estimate the critical exponents no matter in the finite-size scaling region or in the finite-time scaling region. Crossover between the two regions is also demonstrated. We verify our theory using both the one-dimensional transverse-field Ising model and the one-dimensional quantum Potts model. Our results confirm that the Schmidt gap can be regarded as a supplement to the local order parameter., Comment: 7 pages, 7 figures

We present the effect of vacancy in Pr0.6Sr0.4MnO3via dc magnetisation measurements. Using various techniques such as modified Arrott plots, the Kouvel–Fisher method, and Widom scaling relationship the values of TC (ferromagnetic transition temperature), as well as the β, γ and δ (critical exponents) are estimated. Critical exponents for the stoichiometric sample and the strontium deficient sample match well with those predicted for the tricritical mean field model. The vacancy in Pr0.5□0.1Sr0.4MnO3 changes the universal class. The estimated critical exponents of the praseodymium deficient sample are close to those found out by the 3D-Ising model.

The relation between Eulerian structure function’s scaling exponents and Lagrangian ones in turbulent channel flows is explored both theoretically and numerically. A nonlinear parametric transformation between Eulerian structure function’s scaling exponents and Lagrangian ones is derived, following Landau and Novikov’s frame work. This relation is then compared to some known experimental and numerical results, but mainly to our DNS (direct numerical simulation) results of a fully developed channel flow with Reτ = 100. The scaling exponents are evaluated in terms of the ESS (extended self-similarity) method, since the Reynolds number is too low to make the standard scaling laws applicable. The agreement between theory and simulation is satisfactory

A theory of critical phenomena in liquids is constructed based on postulates of classical statistical mechanics (i.e., without invoking hypotheses underlying scaling theory). This theory allows for both long-and short-wave fluctuations, arising in the vicinity of a critical point. Equations for evaluating the parameters of the critical point (critical density, critical temperature, and critical pressure) from the given interaction potential are derived. An equation of state is obtained for the critical isotherm. From this equation it follows that in the vicinity of the critical point, the critical indices have classical values; scaling indices arise away from the critical point.

The general shape of the temperature dependence of the static susceptibility in a biasing field conjugated to the order parameter is analysed with the use of the simplest equation of state compatible with the Widom and Griffiths scaling hypothesis. The corresponding curves are demonstrated to show from two to four inflection points, from which a discontinuous inflection point is found to occur exactly at the critical point whenever the critical exponent of susceptibility differs from one: . The unique inflection point occurring below the temperature of the maximum of the susceptibility in the case of the classical critical exponents, i.e. in the mean field theory, is also shown to be strictly independent of the biasing field. New scaling invariants related to the inflection points are found and their analytical expressions are given for the considered equation of state. The usefulness of the theoretical results to the analysis of experimental data is discussed.

Polycrystalline La0.6Ca0.4MnO3 sample has been prepared by sol-gel method and has an orthorhombic perovskite structure with Pnma space group. Close to magnetic temperature transition, the order transition and the critical behavior are investigated by dc magnetization measurements. Using various techniques such as modified Arrott plots, Kouvel–Fisher method, and Widom scaling relation, the values of T C ferromagnetic transition temperature, as well as the β, γ, and δ critical exponents, are estimated. The values of critical exponents, derived from the magnetic data, yield 0356 ≤ β ≤ 0382, 1273 ≤ γ ≤ 1291, and 4.37 ≤ δ ≤ 457. The critical exponents reported in this study obey the Widom scaling relation δ = 1 + γ/ β, implying that the obtained values of β and γ are reliable. We conclude that La0.6Ca0.4MnO3 belongs to the 3D-Heisenberg class with short-range interaction.

Static magnetization for single crystals of insulating Nd0.85Pb0.15MnO3 and marginally conducting Nd0.70Pb0.30MnO3 has been studied around the ferromagnetic to paramagnetic transition temperature T-C. Results of measurements carried out in the critical range vertical bar(T - T-C)/T-C vertical bar

Widom's particle insertion method is used to calculate effective pair potentials for binary A/B mixtures, where B molecules represent a solvent whose degrees of freedom are integrated out. Comparison of Widom's method with an alternative route via the potential of mean force shows good agreement; however, Widom's particle insertion method experiences sampling difficulties at high densities. An isotherm for a binary mixture of argon (‘solvent’, B) and krypton (‘solute’, A) is calculated. While the structure at the pair level is well represented by the effective system using only effective pair potentials and ignoring all higher multi-body contributions, increasing deviations in the pressure are observed as the density of the effective system is increased. These latter deviations are attributed to the neglect of three- and higher multi-body interactions.

We investigate, for two water models displaying a liquid–liquid critical point, the relation between changes in dynamic and thermodynamic anomalies arising from the presence of the liquid–liquid critical point. We find a correlation between the dynamic crossover and the locus of specific heat maxima \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}C_{P}^{{\mathrm{max}}}\end{equation*}\end{document} (“Widom line”) emanating from the critical point. Our findings are consistent with a possible relation between the previously hypothesized liquid–liquid phase transition and the transition in the dynamics recently observed in neutron scattering experiments on confined water. More generally, we argue that this connection between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}C_{P}^{{\mathrm{max}}}\end{equation*}\end{document} and dynamic crossover is not limited to the case of water, a hydrogen bond network-forming liquid, but is a more general feature of crossing the Widom line. Specifically, we also study the Jagla potential, a spherically symmetric two-scale potential known to possess a liquid–liquid critical point, in which the competition between two liquid structures is generated by repulsive and attractive ramp interactions.

We have extended dynamic scaling hypothesis to describe the wide range of experiments on growing rough interfaces. An alternative approach of original Family–Vicsek scaling hypothesis was not successfully applied to the growing interfaces of paper wetting in that we obtained unphysical values of roughness exponents. Since very nice data collapse suggested the existence of some kind of dynamic scaling, we have tried to extend the dynamic scaling hypothesis. It was found that the extended dynamic scaling hypothesis can be nicely applied to the growing rough interfaces in paper wetting, yielding values of roughness and growth exponents α≃0.73, β≃0.60, respectively.

Using a finite-size phenomenological theory we investigate the behavior of the Baxter–Wu model for both first- and second-order transitions. In order to distinguish between the two kinds of transition we study the finite-size scaling behavior of the order parameter and the susceptibility of the model. At the critical temperature the known critical exponents of the system govern the scaling behavior, while below the critical temperature the scaling exponents are the dimensionality of the system. The presented scaling theory for the Baxter–Wu model is an appropriate generalization of the corresponding theory of Binder and Landau for the Ising model [Binder and Landau, Phys. Rev. B 30 (1984) 1477]. We perform numerical simulations using standard Monte Carlo techniques and we find that our results are in good agreement with theoretical predictions.

Adsorption on small-size one-dimensional clusters is investigated using the Monte Carlo method. The effect of temperature and system size variations on adsorption is studied. Critical exponents of the correlation length and dynamic critical exponent z are calculated taking into account the hypothesis of dynamic scaling. The results obtained demonstrate that non-equilibrium adsorption in nanosystems can occur in a much different fashion than in macrosystems.

The Quantum Hall Plateau-to-Plateau transition is a long-standing problem that has not yet been solved. Around the critical point, very different critical exponents of the localization length have been gotten experimentally. We approach this problem by doing experiments in the alloy disorder, which is realized in the Al x Ga 1-x As - Al 0.32 Ga 0.68 As heterostructure. By measuring directly the alloy scattering rate, we found the alloy disorder to be strong (1.13eV), short-range (atomic size) and temperature-independent3. By controlling the Aluminum concentration x, we make systems dominated by the short-range alloy disorder, where quantum mechanics prevails. In these systems, the scaling exponent is observed to be 0.42.

A systematic way for deriving fundamental earthquake scaling relationships in their size-frequency distribution and high-frequency spectrum falloff based on the renormalization group method is presented for the model earthquakes. It is known that faults populations are broadly scale-invariant over several orders of magnitude. Applying the renormalization group method to the generalized Burrige–Knopoff earthquake model, we obtain the self-similar solution from which the scaling exponents of the size-frequency distribution and spectrum falloff are obtained. The obtained exponents are in agreement with the known values and the relation between these scaling exponents helps to bridge the gap between statistic and dynamical scalings. The present paper permits us to study the self-similar behavior and helps in understanding the basic ingredients underlying earthquake scaling.

A novel method to estimate the critical point and critical exponents of physical models from numerical studies is presented. The method utilizes linear approximation to compute the values of the characteristic variables in the near vicinity of the critical point from numerical results obtained for only one point in that vicinity. The method is applied to two models: Two-dimensional directed percolation, and one-dimensional reaction-diffusion model. In both cases the critical point and critical exponents are determined with higher accuracy than achieved in former studies. In the reaction-diffusion model, the results strongly suggest simple rational values of 1/3, 1/12, etc., for the characteristic exponents.

We propose to combine the nonlinear scaling fields associated with the high-temperature (HT) fixed point, with those associated with the unstable fixed point, in order to calculate the susceptibility and other thermodynamic quantities. The general strategy relies on simple linear relations between the HT scaling fields and the thermodynamic quantities, and the estimation of RG invariants formed out of the two sets of scaling fields. This estimation requires convergent expansions in overlapping domains. If, in addition, the initial values of the scaling fields associated with the unstable fixed point can be calculated from the temperature and the parameters appearing in the microscopic Hamiltonian, one can estimate the critical amplitudes. This strategy has been developed using Dyson's hierarchical model where all the steps can be approximately implemented with good accuracy. We show numerically that for this model (and a simplified version of it), the required overlap apparently occurs, allowing an accurate determination of the critical amplitudes.

Nature Communications 7: Article number: 13611 (2016); Published 5 December 2016; Updated 17 January 2017 The original version of this Article contained a typographical error in the spelling of the author S.A. Cannas, which was incorrectly given as S. Cannas. This has now been corrected in both the PDF and HTML versions of the Article.

Is the brain really operating at a critical point? We study the non-equilibrium properties of a neural network which models the dynamics of the neocortex and argue for optimal quasi-critical dynamics on the Widom line where the correlation length is maximal. We simulate the network and introduce an analytical mean-field approximation, characterize the non-equilibrium phase transition, and present a non-equilibrium phase diagram, which shows that in addition to an ordered and disordered phase, the system exhibits a quasiperiodic phase corresponding to synchronous activity in simulations which may be related to the pathological synchronization associated with epilepsy., Submitted to Physical Review Letters on March 29, 2014. Transferred to Physical Review E on August 4, 2014

The critical behavior of ${\mathrm{Sr}}_{1\ensuremath{-}x}$${\mathrm{Ca}}_{x}$${\mathrm{RuO}}_{3}$ was investigated by a scaling analysis based on the Arrott-Noakes equation of state. The critical exponents \ensuremath{\beta}, \ensuremath{\gamma}, and \ensuremath{\delta} of the magnetic critical behavior were extracted for samples with 0 $\ensuremath{\le}x\ensuremath{\le}0.6$ . The ferromagnetic system exhibits a smooth suppression of the Curie temperature ${T}_{\mathrm{C}}$ to zero at a critical concentration ${x}_{c}$ \ensuremath{\approx} 0.7. The ordered magnetic moment decreases simultaneously as expected for itinerant ferromagnets, however, does not vanish completely at ${x}_{\mathrm{c}}$, indicating small magnetic clusters or inhomogeneities. For $x$ = 0, mean-field like exponents are observed. With increasing $x$, the critical exponents \ensuremath{\beta}, \ensuremath{\gamma}, and \ensuremath{\delta} vary nearly linearly from \ensuremath{\beta} \ensuremath{\approx} 0.5, \ensuremath{\gamma} \ensuremath{\approx} 1, and \ensuremath{\delta} \ensuremath{\approx} 3 for $x$ = 0 to \ensuremath{\beta} \ensuremath{\approx} 1, \ensuremath{\gamma} \ensuremath{\approx} 0.9, and \ensuremath{\delta} \ensuremath{\approx} 1.6 for $x$ \ensuremath{\approx} 0.6. The Widom scaling relation is always met for $x\ensuremath{\le}0.6.$ Despite the systematic evolution of the critical exponents as a function of $x$, the exponents cannot be described by any of the universality classes known for classical standard models. The particular trend of the effective critical exponents may be possibly explained by a strong-disorder line of fixed points; the vicinity to ${x}_{c}$ suggests that this behavior is caused by the vicinity to a quantum phase transition.

Scaling analysis, in which one infers scaling exponents and a scaling function in a scaling law from given data, is a powerful tool for determining universal properties of critical phenomena in many fields of science. However, there are corrections to scaling in many cases, and then the inference problem becomes ill-posed by an uncontrollable irrelevant scaling variable. We propose a new kernel method based on Gaussian process regression to fix this problem generally. We test the performance of the new kernel method for some example cases. In all cases, when the precision of the example data increases, inference results of the new kernel method correctly converge. Because there is no limitation in the new kernel method for the scaling function even with corrections to scaling, unlike in the conventional method, the new kernel method can be widely applied to real data in critical phenomena., Comment: 8 pages, 5 figures, the reference code of this new method is prepared at http://kenjiharada.github.io/BSA/

The (La1−xNdx)2/3(Ca1−ySry)1/3MnO3 have been elaborated by the solid-state method and checked by X-ray diffraction. Close to magnetic temperature transition, the order transition and the critical behavior are investigated by dc magnetization measurements versus x and y compositions. Using various techniques such as modified Arrott plots, Kouvel–Fisher method, and Widom scaling relation, the values of TC ferromagnetic transition temperature, as well as the β, γ and δ critical exponents are estimated. The scaling plots show that the obtained values of the critical exponents are reliable. As main results, metamagnetic behavior and Griffiths phase were observed in these compounds pointing out to unconventional critical exponent.

Phase diagrams of the SO(5) theory are calculated by Monte Carlo simulations. A scaling theory is developed for the bicritical phenomenon and applied to analyze the critical phenomena of the superfluid density. It is then proposed to perform systematic experimental measurements on the London penetration depth of high- T c cuprates from the optimal to underdoped regime. By applying the scaling analysis, it is possible to estimate the critical exponents of the penetration depth with high accuracy for all doping rates in the phase diagram. From both the critical exponents and the scaling function, one can tell whether the degrees of freedom is really five as in the SO(5) theory or not. A tricritical point on the normal to antiferromagnetism phase boundary is predicted when an external magnetic field is applied.

Asymptotic and leading correction to scaling critical exponents and amplitudes have been determined for quenched amorphous Fe 90− y Mn y Zr 10 ( y =0–8) ferromagnets through an elaborate analysis of temperature dependence of spontaneous magnetization, zero-field susceptibility and low-field AC susceptibility data obtained in the asymptotic critical region. From this analysis, it is found that the values of the critical exponents and amplitudes do not depend on the alloy composition and are in good agreement with the values predicted for three-dimensional Heisenberg ferromagnet system. The observed experimental results are consistent with the concept of scaling in that the exponent equalities β = γ ( δ −1) and α =2(1− β )− γ are obeyed to a high degree of accuracy. These results show that both amorphous and crystalline materials behave similarly in the critical region though amorphous alloys show a wide asymptotic critical region than the crystalline materials. The presence of disorder does not seem to have any influence on critical behavior of the system investigated in the present work.

We report on extensive numerical simulations of the three-dimensional Heisenberg model and its analysis through finite-size scaling of Lee-Yang zeros. Besides the critical regime, we also investigate scaling in the ferromagnetic phase. We show that, in this case of broken symmetry, the corrections to scaling contain information on the Goldstone modes. We present a comprehensive Lee-Yang analysis, including the density of zeros and confirm recent numerical estimates for critical exponents., Comment: 19 pages, 9 figures

ac ("zero-field") susceptibility of the amorphous ferromagnetic alloys (Fe p Ni 1-p ) 80 B 16 Si 4 (0.0540 p' Ni 1-p' ) 80 B 16 Si 4 (0.1125 c or p c' for the onset of long-range ferromagnetic order has been measured to very high precision in the absence and presence of dc (static) magnetic field over a wide temperature range embracing the critical region near the ferromagnetic-paramagnetic phase transition. Elaborate data analyses permit the accurate determination of the asymptotic critical exponents β D , γ D , and δ D for spontaneous magnetization, susceptibility and the critical isotherm as well as the leading "correction-to-scaling" exponent for susceptibility (these exponents characterize the isotropic dipolar fixed point) and hence assert that the asymptotic critical behavior of amorphous ferromagnets with p c ≲p c '≲p' D , γ D , and δ D do satisfy the Widom scaling relation β+γ=βδ accurately. In addition, the temperature dependence of the effective critical exponent for susceptibility observed in the amorphous ferromagnetic alloys with p C as e=(T-T C )/T C ≃ 10 -5 in the alloys with p>0.1 or p'>0.2, which behave as normal ferromagnets down to 3.8 K. A sharp contrast in the critical behavior of ferromagnets that either do or do not exhibit reentrant behavior at low temperatures is shown to reflect the decisive role played by the isotropic dipolar long-range interactions in establishing long-range ferromagnetic order in dilute magnetic systems exhibiting reentrant behavior.

We use the finite size scaling method to study the critical points, points of non-analyticity, of the ground state energy as a function of the coupling parameters in the Hamiltonian. In this approach, the finite size corresponds to the number of elements in a complete basis set used to expand the exact eigenfunction of a given molecular Hamiltonian. To illustrate this approach, we give detailed calculations for systems of one electron and two nuclear centres, Z + e −Z+. Within the Born-Oppenheimer approximation, there is no critical point, but without the approximation the system exhibits a critical point at Z = Zc = 1.228279 when the nuclear charge, Z, varies. We show also that the dissociation occurs in a first-order phase transition and calculate the various related critical exponents. The possibility of generalizing this approach to larger molecular systems using Gaussian basis sets is discussed.

We investigate a method to obtain critical exponents using standard numerical simulation techniques for spin systems. We show that by analyzing the inverse of the logarithmic derivative of the susceptibility χ with respect to temperature the accuracy in determining γ can be increased significantly. The method is tested on the two- and three-dimensional Ising models but this alternative approach is thought to be most useful for spin systems on non-periodic structures like quasi-crystals and fractal lattices where the finite size scaling method is difficult to apply.

The six-dimensional Ising model is simulated on the Creutz cellular automaton using the finite-size lattices with the linear dimension 4 less than or equal to L less than or equal to 10. The exponents in the finite-size scaling relations for the magnetic susceptibility and the order parameter at the infinite-lattice critical temperature are computed to be 3.00(12) and 1.47(10) using 6 less than or equal to L less than or equal to 10, respectively, which are in very good agreement with the theoretical predictions of 6/2 and a. The critical temperature for the infinite lattice is found to be 10.835(5) using 4 less than or equal to L less than or equal to 10 which is also in very good agreement with the precise results. The finite-size scaling relation fur the magnetic susceptibility at the infinite-lattice critical temperature is also valid for the maxima of the magnetic susceptibilities of the finite-size lattices. The finite-size scaling plots of the magnetic susceptibility and the order parameter verify the finite-size scaling relations about the infinite-lattice critical temperature. (C) 2000 Elsevier Science B.V. All rights reserved.

The surface scaling behavior of classical isotropic Heisenberg magnets is investigated by Monte - Carlo methods in d=3 dimensions for various values of the surface - to - bulk coupling ratio J_1/J. For J_1/J = 2.0 scaling is still valid with effective exponents which depend on J_1/J. The surface structure factor S_1(p,L) is investigated at bulk criticality as function of the momentum transfer p parallel to the surface and the system size L. For J_1/J = 2.0 the full p dependence of S_1(p,L) can be captured by generalized shape functions to a remarkable accuracy. Profiles of the magnetization and the energy density also confirm scaling, where for J_1/J = 2.0 scaling with J_1/J dependent exponents is found. For J_1/J = 1.5 the system displays a striking crossover behavior from spurious long - range surface order to the ordinary surface universality class. For J_1/J >= 2.0 the effective scaling laws must be interpreted as nonasymptotic and the value J_1/J = 1.5 marks a crossover regime, in which the crossover from the nonasymptotic to the asymptotic (ordinary) surface scaling behavior can be resolved within numerically attainable system sizes., 14 pages RevTeX, 14 figures; to appear in Phys. Rev. B, Sept. 2000

The renormalization group theory is applied to the determination of localized phonons in a finite size two dimensional monolayer. The fixed-point equation defines the critical values of the adsorbate-substrate force constants leading to the occurrence of localized modes. The transformation equation near the fixed point gives scaling laws and critical exponents which characterize the asymptotic behavior of the localization length of the phonons.

The bond-percolation process is studied on periodic planar random lattices and their duals. The thresholds and critical exponents of the percolation transition are determined. The scaling functions of the percolating probability, the existence probability of the appearance of percolating clusters, and the mean cluster size are also calculated. The simulation result of the percolation threshold is pc50.333360.0001 for planar random lattices, and 0.667060.0001 for the duals of planar random lattices. We conjecture that the exact value of pc is 1/3 for a planar random lattice and 2/3 for the dual of a planar random lattice. By taking possible errors into account, the results of our critical exponents agree with the values given by the universality hypothesis. By properly adjusting the metric factors on random lattices and their duals, we demonstrate explicitly that the idea of a universal scaling function with nonuniversal metric factors in the finite-size scaling theory can be extended to random lattices and their duals for the existence probability, the percolating probability, and the mean cluster size. @S1063-651X~99!00112-9#

A mean-field free-energy functional for an n-component mixture with an integral non-local interaction is introduced and then written explicitly for a binary mixture. We use this functional to calculate the liquid–vapor surface tension with parameters chosen to model CO2/n–C4H10 and CO2/n–C10H22, and we examine the scaling of the surface tension as a function of the difference in density between the liquid and vapor phases as various critical points are approached. Each critical point is approached on either a constant-temperature or constant-pressure path; we investigate the path dependence of the scaling behavior. For the constant-temperature paths in the CO2/n–C4H10 mixture, we compare our calculated results with experimental data. We find no significant dependence of the scaling on the path to the critical point. We note that the asymptotic scaling holds for a larger range of densities the higher the temperature of the critical point.

The static critical properties of the Cr2O3 real antiferromagnetic model are investigated by the Monte-Carlo method. The systems with periodic boundary conditions containing N=500 ; 864; 1372; 2048; 2916; 4000 spins are studied. Using the theory of finite-size scaling, the values of critical exponents α , β , γ are calculated.The comparison of data with the results of theoretical and experimental research is carried out.

We study the critical scaling behaviors of period doublings in N (N=2,3,4,…) coupled magnetic oscillators by varying the driving amplitude A and the coupling strength c. It is found that the critical scaling behaviors depend on the range of coupling. For the extreme long-range case of global coupling, the critical set (set of critical points) is composed of the zero-coupling critical point with c=0 and an infinity of critical line segments in the A-c plane, independently of N. Three kinds of critical scaling behaviors are found on the critical set. However, for any other nonglobal-coupling cases of shorter-range couplings, the structure of the critical set becomes different from that for the global-coupling case, because of a significant change in the stability diagram. We also note that the structure of the critical set and the critical scaling behaviors for both cases of the global and nonglobal couplings are the same as those in the abstract system of the coupled one-dimensional maps.