Griffiths Model Research Articles

Critical and reentrant phenomena in the Blume–Emery–Griffiths model with attractive biquadratic interaction

Critical and reentrant phenomena in the Blume–Emery–Griffiths model with attractive biquadratic interaction

Monte Carlo and mean-field studies on the magnetic properties of a hexagonal Ising nanowire with core-shell structure in the Blume–Emery–Griffiths model

The mean-field approximation based on the Gibbs-Bogoliubov inequality and Monte Carlo simulations based on the Metropolis method in the Blume–Emery–Griffiths model were used to successfully examine a hexagonal Ising nanowire with spin-1. It was found that the model displays fascinating phase diagrams in various planes of interest by using both methods. The investigation of magnetic hysteresis cycles also exhibits interesting properties. The model not only presents first- and second-order phase transition lines, but also the tricritical and isolated critical points, multi-compensation temperatures, and magnetic hysteresis behavior with one to eight cycles.

Blume–Emery–Griffiths model for mixed spin (1, 5/2) calculated by mean-field approximation

Blume–Emery–Griffiths model for mixed spin (1, 5/2) calculated by mean-field approximation

The Blume–Emery–Griffiths Model on the FAD Point and on the AD Line

We analyse the Blume–Emery–Griffiths (BEG) model on the lattice Zd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}^d$$\\end{document} on the ferromagnetic-antiquadrupolar-disordered (FAD) point and on the antiquadrupolar-disordered (AD) line. In our analysis on the FAD point, we introduce a Gibbs sampler of the ground states at zero temperature, and we exploit it in two different ways: first, we perform via perfect sampling an empirical evaluation of the spontaneous magnetization at zero temperature, finding a non-zero value in d=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=3$$\\end{document} and a vanishing value in d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document}. Second, using a careful coupling with the Bernoulli site percolation model in d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document}, we prove rigorously that under imposing +\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$+$$\\end{document} boundary conditions, the magnetization in the center of a square box tends to zero in the thermodynamical limit and the two-point correlations decay exponentially. Also, using again a coupling argument, we show that there exists a unique zero-temperature infinite-volume Gibbs measure for the BEG. In our analysis of the AD line we restrict ourselves to d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document} and, by comparing the BEG model with a Bernoulli site percolation in a matching graph of Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}^2$$\\end{document}, we get a condition for the vanishing of the infinite-volume limit magnetization improving, for low temperatures, earlier results obtained via expansion techniques.

The microcanonical and the canonical phase diagrams of the mean-field Blume–Emery–Griffiths model

The microcanonical and the canonical phase diagrams of the mean-field Blume–Emery–Griffiths model

Critical and hysteresis behaviors for a hexagonal core-shell structure nanowire in the Blume–Emery–Griffiths model

The magnetic properties and hysteresis loops for the hexagonal Ising nanowire (HIN) with core–shell structure consisting of spin-32 are considered in the Blume–Emery–Griffiths (BEG) model by using the mean-field approximation (MFA) based on the Gibbs–Bogoliubov inequality for the free energy. The effects of various bilinear and biquadratic interaction parameters between the core, between the shell, and between the core and shell spins are considered for the phase diagrams of the model when temperature T=0 and T≠0 in addition to the crystal and external magnetic fields effects. The numerical calculations reveal that the model yields first- and second-order phase transition lines and tricritical points. In addition to the isolated critical and critical end-points, the model exhibits at most two compensation temperatures and interesting multiple hysteresis magnetic loops behaviors strongly dependent on the model parameters.

Critical behaviour near critical end points and tricritical points in disordered spin-1 ferromagnets

Critical end points and tricritical points are multicritical points that separate lines of continuous transitions from lines of first order transitions in the phase diagram of many systems. In models like the spin-1 disordered Blume–Capel model and the repulsive Blume–Emery–Griffiths model, the tricritical point splits into a critical end point and a bicritical end point with an increase in disorder and repulsive coupling strength respectively. In order to make a distinction between these two multicritical points, we investigate and contrast the behaviour of the first order phase boundary and the co-existence diameter around them.

Geometrical aspects of the multicritical phase diagrams for the Blume–Emery–Griffiths model

As a continuation of our preceding work [R. Erdem and N. Alata, Eur. Phys. J. Plus 135, 911 (2020), https://doi.org/10.1140/epjp/s13360-020-00934-3], we used the thermodynamic geometry in the Ruppeiner formalism to study the geometrical aspects of the multicritical phase diagrams for the spin-1 Blume-Emery- Griffiths model in the presence of crystal field. We derived an expression for the thermodynamic curvature or Ricci scalar (R) and analyzed its temperature and crystal field behaviours near interesting critical and multicritical points. Our findings are presented as geometrical phase diagrams including critical and multicritical topology. From these diagrams, new vanishing curvature lines (R = 0) extending into the ferromagnetic or paramagnetic phases beyond the critical points and zero point temperature are observed.

Gibbs Measures of the Blume–Emery–Griffiths Model on the Cayley Tree

Gibbs Measures of the Blume–Emery–Griffiths Model on the Cayley Tree

Blume–Emery–Griffiths model on random graphs

The Blume–Emery–Griffiths model with a random crystal field is studied in a random graph architecture, in which the average connectivity is a controllable parameter. The disordered average over the graph realizations is treated by replica symmetry formalism of order parameter functions. A self consistent equation for the distribution of local fields is derived, and numerically solved by a population dynamics algorithm. The results show that the average connectivity amounts to changes in the topology of the phase diagrams. Phase diagrams for representative values of the model parameters are compared with those obtained for fully connected mean field and renormalization group approaches.

Study of the Blume–Emery–Griffiths Model for Mixed Carbon-Like Nanotube: Monte Carlo Study

Monte Carlo study under the Metropolis algorithm is performed to investigate the ground state phase diagrams and hysteresis electric cycles by using the Blume–Emery–Griffiths (BEG) model with the mixed spins (S-1, σ-3/2). Firstly, the ground state phase diagram has been established to show the more stable configurations corresponding to the physical parameter EZ/JC. Moreover, the hysteresis electric cycle behaviors have been investigated by varying temperature, exchange coupling interactions, crystal field and biquadratic parameters of the carbon-like nanotube. It is found that the increase in the crystal field and the biquadratic exchange parameters decrease the surface of the loops leading to the apparition of the polarization plateaus.

Thermodynamic geometry of spin-one lattice models. I. Spin and quadrupolar orders and critical scaling functions in one dimension.

State space Riemannian geometry is obtained for the one-dimensional Blume-Emery-Griffiths model and its Blume-Capel and Griffiths model limits, and its (pseudo)critical as well as noncritical parameter regimes are extensively investigated. Two codimension one geometries are obtained by taking suitable hypersurfaces in the three-dimensional state space manifold, and the induced thermal metrics are accordingly interpreted in terms of constrained fluctuations. The three-dimensional scalar curvature and the two two-dimensional curvatures are shown to be consistent with Ruppeiner's conjecture relating the inverse of the singular free energy to the thermodynamic scalar curvature. Moreover, they are found to be in an excellent agreement over a greater part of the noncritical region with the corresponding correlation lengths for the spin and the quadrupolar order parameters. The scaling function for the free energy near the pseudocritical and tricritical points is obtained thermodynamically by using Ruppeiner's conjecture. A connection is made between the sign change in the curvatures and the change in fluctuation patterns of the order parameters. In the accompanying paper we shall analyze the geometry of the spin-one model in its mean field approximation.

Blume–Emery–Griffiths model for a magnetic bilayer. Magnetocaloric properties in the presence of the 1st-order phase transitions

In the paper the Blume–Emery–Griffiths model is studied for a magnetic bilayer with coordination number z=5. The Cluster Variational Method in Pair Approximation is employed. A special emphasis is put on the magnetic and magnetocaloric properties in the cases of the 1st order phase transitions. For such cases the magnetization, quadrupolar moment and magnetic entropy, as well as the isothermal entropy change and the adiabatic temperature change during demagnetization process are calculated. It is shown that in the vicinity of the temperature corresponding to the 1st order phase transitions the adiabatic demagnetization is not possible. Instead, the demagnetization process with the smallest entropy production is proposed.

Dynamic multicritical phase diagrams of the mixed spin (2, 5/2) Blume-Emery-Griffiths model with repulsive biquadratic coupling

We investigated the dynamic phase transitions (DPTs) in the mixed spin (2, 5/2) Blume-Emery Griffiths model with repulsive biquadratic interaction in the presence of a time-varying magnetic field. We used the path probability method to obtain the set of the dynamic equations. We numerically solved these dynamic equations to characterize the nature of first- and second-order phase transitions and to find the DPT temperatures as well as obtain the phases in the system. We constructed the dynamic phase diagrams (DPDs) in reduced temperature and amplitude of oscillating magnetic field plane. We observed that the DPDs display richer, complex and more topological various type of phase diagrams. In particular, DPDs exhibit the disordered phase, antiquadrupolar or staggered phase, six different ferrimagnetic phases, three different nonmagnetic phases, and numerous mixed phases. DPDs also display two dynamic tricritical points for only smaller values of crystal-field interactions, multiple critical end and double critical end points, one zero-temperature critical point, one inverse critical end point, and a quadruple point depending on interaction parameters. The system always shows the reentrant behaviors for the higher values of magnetic field amplitude, but it does not exhibit the dynamic tricritical behavior for higher values of crystal-field parameter.

Electrostatic Conditions That Produce Fast Breakdown in Thunderstorms

AbstractFast breakdown (FB), a breakdown process composed of systems of high‐velocity streamers, has been observed to precede lightning leader formation and play a critical role in lightning initiation. Vigorous FB events are responsible for the most powerful natural radio emissions on Earth, known as narrow bipolar events (NBEs). In this paper, an improved version of the Griffiths and Phelps (1976, https://doi.org/10.1029/jc081i021p03671) model of streamer breakdown is used alongside supervised machine learning techniques to probe the required electric fields and potentials inside thunderstorms to produce FB and NBEs. Our results show that the electrostatic conditions needed to produce FB observed in New Mexico at 9 km altitude and FB in Florida at 14 km altitude are about the same, each requiring about 100 MV potential difference to propagate 500 m. Additionally, the model illustrates how electric field enhancement ahead of propagating FB can initiate rebounding FB of the opposite polarity.

Hysteretic dipolar and quadrupolar properties of the spin-1 Blume–Emery–Griffiths model under pair approximation

As a continuation of previously published work [A. Erdinç and M. Keskin, Int. J. Mod. Phys. B 18, 1603 (2004), doi:10.1142/S0217979204024823] the hysteretic properties of magnetization ([Formula: see text]) and quadrupolar moment ([Formula: see text] have been investigated for the spin-1 Blume–Emery–Griffiths model using pair approximation. The magnetic field ([Formula: see text]) dependence of [Formula: see text] is produced for different values of the lattice coordination number ([Formula: see text]), biquadratic interaction ([Formula: see text]) and single-ion anisotropy ([Formula: see text]). The ([Formula: see text]) loops show different and novelty properties when [Formula: see text]. We also observed the hysteresis cycles in the ([Formula: see text], ([Formula: see text] and ([Formula: see text]) planes. These curves strongly depend on biquadratic interaction [Formula: see text]. Our results are found to be in good agreement with the other theoretical findings.

Phase diagram of the repulsive Blume–Emery–Griffiths model in the presence of an external magnetic field on a complete graph

Using the repulsive Blume–Emery–Griffiths model, we compute the phase diagram in three field spaces, temperature (T), crystal field (Δ), and magnetic field (H) on a complete graph in the canonical and microcanonical ensembles. For low biquadratic interaction strengths (K), a tricritical point exists in the phase diagram where three critical lines meet. As K decreases below a threshold value (which is ensemble dependent), new multicritical points such as the critical end point and the bicritical end point arise in the (T, Δ) plane. For K > −1, we observe that the two critical lines in the H plane and the multicritical points are different in the two ensembles. At K = −1, the two critical lines in the H plane disappear, and as K decreases further, there is no phase transition in the H plane. At exactly K = −1, the two ensembles become equivalent. Beyond that, for all K < −1, there are no multicritical points, and there is no ensemble inequivalence in the phase diagram. We also study the transition lines in the H plane for positive K, i.e. for attractive biquadratic interaction. We find that the transition lines in the H plane are not monotonic in temperature for large positive values of K.

Ruppeiner geometry of isotropic Blume–Emery–Griffiths model

With the aid of Ruppeiner thermodynamic metric defined on a two-dimensional phase space of dipolar (m) and quadrupolar (q) order parameters, we derive an expression for the Ricci scalar (R) in the isotropic Blume–Emery–Griffiths model. Temperature dependence of R is investigated for various values of bilinear to biquadratic ratio (r). Its behavior near the continuous/discontinuous phase transition temperatures and a tricritical point is presented. It is found that in addition to the divergence singularity and finite jumps connected with the phase transitions, there are field-dependent broad extrema in the Ricci scalar.

Griffiths phase-like exponents in the hidden-order state of URu2Si2

We report electrical resistivity, magnetic susceptibility, and specific heat measurements of high-quality URu2Si2 single crystals at ambient pressure, in the hidden-order (HO) region of the phase diagram. An amazing power-law behavior is observed, for both DC susceptibility and specific heat, establishing the Griffiths model for the HO phase. The Griffiths phase is characterized by residual short-range correlations on the collapse of the long-range large-moment antiferromagnetic (LMAFM) phase due to the dilution effects. The existence of cluster-like spins is intrinsic, as evidenced by the frequency dispersion of AC susceptibility and the relaxation of magnetoresistance. On account of the freezing of magnetic clusters, an unidirectional anisotropy of the resistivity in rotating magnetic fields is detected, which naturally explains any possible broken symmetries in tiny and clean crystals. In this manner, the HO phase of URu2Si2 is intimately related to the LMAFM phase. The proposal of Griffiths phase as an alternative of the HO phase is very promising, which may open a new route into the understanding of exotic electronic states in correlated matter.

Crack growth under three-point beam bending and fracture energy estimation for snow in relation to acoustic emission responses

The non-invasive monitoring of crack initiation and growth processes within snow, a poro-composite matrix, could be useful to predict the structural failure of a composite material. Present work is a new approach utilizing the non-invasive technique of acoustic emission (AE) to study the ongoing cracking and failures processes within snow beams which is difficult to visualize in real time. The progressive crack growth within the beams of snow followed by their failures is studied under three point beam bending in relation to AE. In this work, AE detection platform is developed and implemented for non-contact detection of acoustic emissions generated within a beam with the help of specialized needle-type AE waveguides in conjunction to wideband AE acquisition system. The AEs are detected for entire course of bending, crack nucleation, crack growth and rupture of the beams, both under static and dynamic load conditions. The crack initiation and growth in snow beams were generated without any predefined notch on its surface in a controlled and perceptible manner, particularly at low deformation rates (<0.2 mm/min). The ongoing crack growth, in terms of crack lengths and openings, were also studied from high resolution images, regularly collected during the experimental course. The prominent AE characteristics such as counts, amplitude, absolute energy, etc. were analyzed in context of bending, cracking and rupture of the snow beams for static as well as dynamic load conditions. The total strain energy, estimated in terms of the strain energy density, was imparted to the snow beam which contributed in the crack initiation and growth processes. Further, a fraction of the crack surface energy was manifested in the form of AE energy; therefore, a correlation was established between the crack surface energy and the liberated AE energy for static as well as dynamic loading of snow. The Griffiths model for global energy balance was applied to estimate the crack surface energy. The relation between crack surface energy and corresponding AE energy (released) could be very useful to estimate the fracture energy of the materials directly in terms of AE, non-invasively.