Blume-Emery-Griffiths Research Articles

Thermodynamic information geometry of criticality, fluctuation anomaly and hyperscaling breakdown in the spin-3/2 chain

Equilibrium state space geometry is worked out for the exactly solved one-dimensional spin-3/2 lattice with a Blume–Emery–Griffiths (BEG) Hamiltonian as well as a more general one with a non-zero field coupled to the octopole moments. The phase behaviour of the spin-3/2 chain is revisited and novel observations of anomaly in the hyperscaling relation and in the decay of fluctuations are reported for the first time. Using the method of constrained fluctuations developed earlier in Sahay (2017); Sanwari and Sahay (2022) three appropriate state space sectional curvatures and a 3d curvature are shown to separately encode dipolar, quadrupolar and octopolar correlations both near and away from pseudo-criticality. In all observations of hyperscaling breakdown the 3d scalar curvature is found to encode the correlation length while the relevant sectional curvature equals the inverse of singular free energy. For parameter values where the order parameter fluctuation anomalously decays despite a divergence in its correlation length the relevant scalar curvature undergoes a sign change to positive values, signalling a possible change in the nature of the underlying statistical interactions.

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.

Ensemble dependence of the critical behavior of a system with long-range interaction and quenched randomness.

We propose a hybrid model governed by the Blume-Emery-Griffiths (BEG) Hamiltonian with a mean-field-like interaction, where the spins are randomly quenched such that some of them are "pure" Ising and the others admit the BEG set of states. It is found, by varying the concentration of the Ising spins, that the model displays different phase portraits in concentration-temperature parameter space, within the canonical and the microcanonical ensembles. Phenomenological indications that these portraits are rich and rather unusual are provided.

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.

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.

Effects of the random single-ion anisotropy on the spin-1 Blume-Emery-Griffiths model

The Blume-Emery-Griffiths (BEG) model is considered on the Bethe lattice (BL) in terms of exact recursion relations (ERR) under the effect of a crystal field (D) which was either turned on or off randomly for a given probability. The repulsive case of biquadratic exchange interaction ((K < 0)) between the nearest-neighbor (NN) spins is assumed and the effects of changing the coordination number are also investigated. The thermal variation of the order-parameters, i.e. dipole and quadrupole moments, is examined to obtain the phase diagrams. Very rich phase diagrams with the second- and first-order phase transition lines, tricritical, bicritical and critical end points and the occurrence of reentrant behavior are observed. Different phase regions were explored which include the ferromagnetic (F), paramagnetic (P), staggered quadrupolar (SQ) and ferrimagnetic (FI) phases.

Magnetic-field induced multi-step transitions in ferromagnetic spin-crossover solids within the BEG model

We study by means of the 2D Blume–Emery–Griffiths (BEG) spin-1 model, spin-crossover (SCO) and prussian blue analogs (PBAs) solids. In this model, the spin states, which can be high-spin (HS) or low-spin (LS), interact magnetically and elastically with their nearest neighbors. To account for the volume change, accompanying the spin transition phenomenon, all interactions through the lattice are assumed as temperature-dependent. In addition, the system is subject to a variable magnetic field lifting the degeneracy in the HS state. A stochastic cooperative dynamics of this BEG-like Hamiltonian, describing the equilibrium and nonequilibrium properties of ferromagnetic spin-crossover solids, is derived from the Glauber approach, with appropriate Arrhenius microscopic transition rates. The model generates under the magnetic field, sigmoidal relaxation and a hysteresis phenomenon of the HS fraction, as well as multistep behavior of the magnetization. These behaviors open the way to new route of multi-stable systems, desired in multi-byte electronics.

Staggered Quadrupolar Phase in the Bond-Diluted Spin-1 Blume-Emery-Griffiths Model

The random bond-dilution effects of bilinear interaction parameter Jij between the nearest-neighbor (NN) sites are taken into consideration for the spin-1 Blume-Emery-Griffiths (BEG) model on the Bethe lattice (BL) comprised of two interpenetrating equivalent sublattices A and B for given coordination number z in terms of exact recursion relations (ERR). A bimodal distribution for Jij is assumed which is either introduced with probability p or closed with 1 − p. It is assumed that the biquadratic exchange interaction parameter (K) is constant between the NN spins and the single-ion anisotropy parameter (D) is taken to be equivalent on the sublattices A and B. After the study of thermal changes of the order-parameters, the phase diagrams are calculated on possible planes spanned by our system parameters. It is found that the model presents both first- and second-order phase transitions. In addition to the well-known ferromagnetic (F), paramagnetic (P) and ferrimagnetic (FI) phases, the staggered quadrupolar (SQ) phase is also observed. The bicritical point (BCP) for all z and double BCP with z ≥ 4 are observed. The tetracritical point was also found for lower values of p with z ≥ 5.

Monte Carlo simulations of hysteresis effects at the martensitic transformation

In this work, we present a theoretical study of the thermal hysteresis that takes place across the first order martensitic transition as a function of temperature for Heusler alloys. To simulate the structural phase transitions, we propose the degenerate three-state Blume-Emery-Griffiths (BEG) Hamiltonian taking into account the temperature-dependent lattice entropy expansion in the framework of quasi-harmonic Debye model. It is shown that the hysteresis width of structural transformation between cubic austenite and tetragonal martensite depends on the difference between σTM and σTA variables.

Thermodynamics of quantum lattice system with local multi-well potentials: dipole ordering and strain effects in modified Blume–Emery–Griffiths model

ABSTRACTSome aspects of thermodynamics of the quantum lattice model with the local anharmonic potentials are considered for the case of deformed lattice. The effects, taking place under external pressure, are investigated in the framework of the Blume–Emery–Griffiths (BEG) model corresponding to the local potential with three minima. The deformable BEG model is proposed for this purpose; it is taken into consideration that the influence of mechanical stress is realized through the lattice strain resulting in restructuring of local atomic configurations. Based on this model, the pressure dependences of the volume deformation are calculated on an example of the SnPS crystal. The presence of anomalies of function in the regions of ferroelectric phase transitions of the first and second order as well as the tricritical point is established; the behaviour of the volume compressibility in these cases is investigated. Obtained results are in agreement with the experiment.

Dynamical Properties of Spin‐Crossover Solids Within the Kinetic Spin‐1 BEG Model in the Presence of a Time‐Dependent Magnetic Field

AbstractSpin‐crossover (SCO) and Prussian blue analogs (PBAs) materials are investigated in 2D with a three‐state Blume–Emery–Griffiths (BEG) model where each spin interacts with its nearest neighbors () and may be either in high‐spin (HS) or low‐spin (LS) state. The interactions through the system lattice are temperature‐dependent to account for spin‐phonon interactions. The system is also in contact with an oscillating magnetic field energy. The generated numerical results by the dynamic mean field theory (DMFT) study approach are consistent with those derived by kinetic Monte Carlo (KMC) simulations with Glauber dynamics and Arrhenius transition rates. First‐order transitions with thermally induced hysteresis phenomena have been observed. Near the hysteresis loops, the model exhibits throughout relaxation curves, some fluctuations in the LS phase, strengthened by increasing temperature where this phenomenon becomes temperature‐ and magnetic field‐dependent.

Mixed spin-1 and spin-3/2 Blume-Emery-Griffiths model with external field on a honeycomb lattice

In the present work, we use Monte Carlo simulations (MCS) to study the ferrimagnetic of the mixed spin-1 and 3/2 Blume-Emery-Griffiths (BEG) model on a honeycomb lattice. The ground state phase diagram has been obtained for T = 0 whereas at non vanishing, total and partial magnetizations have been calculated. Moroever, we have presented the results of phase diagrams depending on different parameters: exchange coupling paramaters (r1 and r2), crystal field (d1 and d2), biquadratic coupling interaction (k) and external magnetic field (h). On the other hand, the compensation temperature and hysterisis cycle have been established.

The bimodal random crystal field and biquadratic exchange interaction effects for the spin-3/2 Ising model on the Bethe lattice

The phase transition properties of Blume–Emery–Griffiths (BEG) model for the spin-3/2 system are investigated on the Bethe lattice (BL) when the system is under the effect of both random crystal field (D) and biquadratic exchange interaction (K). These randomization effects are either turned on with probability 1−p (q) or turned off with probability p (1−q) for D and K, respectively. The phase diagrams are obtained on the (K/J, kT/J) and (D/J, kT/J) planes for given values of p and q when z=3.0 corresponding to honeycomb lattice. Both attractive (K > 0) and repulsive (K < 0) biquadratic exchange interaction values are considered to examine its effects on the BL. It is found that the model presents either second- or first-order phase transitions and also tricritical points. It is also found that the second-order phase lines follow the phase lines of regular spin-3/2 BEG model as K → ± ∞ for the phase diagrams on the (K/J, kT/J) planes.

Relaxation dynamics and growth modes of metastable spin-crossover materials with magnetic interactions

A two-dimensional spin-1 Blume-Emery-Griffiths (BEG) model is used to describe the relaxation dynamics of spin-crossover (SC) compounds with magnetic interactions from a metastable high-spin state by means of Monte Carlo simulations with Arrhenius dynamics. The model comprises temperature-dependent effective spin exchange interaction accounting for spin-phonon interactions. The growth modes of the stable low-spin state domains are singled out in the model parameters' space. For weak magnetic interactions, numerical results indicated a kind of 2D-nucleation with the birth, spread and coalescence of low-spin diamagnetic domains following the Becker-Doering law. A schematic growth phase diagram is presented.

Comparison of the ferromagnetic Blume–Emery–Griffiths model and the AF spin-1 longitudinal Ising model at low temperature

We derive the exact Helmholtz free energy (HFE) of the standard and staggered one-dimensional Blume–Emery–Griffiths (BEG) model in the presence of an external longitudinal magnetic field. We discuss in detail the thermodynamic behavior of the ferromagnetic version of the model, which exhibits magnetic field-dependent plateaux in the z-component of its magnetization at low temperatures. We also study the behavior of its specific heat and entropy, both per site, at finite temperature. The degeneracy of the ground state, at T=0, along the lines that separate distinct phases in the phase diagram of the ferromagnetic BEG model is calculated, extending the study of the phase diagram of the spin-1 antiferromagnetic (AF) Ising model in S.M. de Souza and M.T. Thomaz, J. Magn. and Magn. Mater. 354 (2014) 205 [5]. We explore the implications of the equality of phase diagrams, at T=0, of the ferromagnetic BEG model with K|J|=−2 and of the spin-1 AF Ising model for D|J|>12.

Ferroelectricity in (PbySn1−y)2P2S6 mixed crystals and random field BEG model

AbstractIn SnPS ferroelectric under compression, the second‐order phase transition line is observed down to the tricritical point as the transition temperature decreases to 250 K. A tricriticality at a similar temperature is also revealed in the mixed crystals on SSe substitution. For the mixed crystals with SnPb substitution, ultrasound, hypersound, and low‐frequency dielectric studies also show heterophase features appearing on decrease of the ferroelectric transition temperature below a so‐called “waterline temperature” near 250 K. Such behavior agrees with the Blume–Emery–Griffiths (BEG) model appropriate for the ferroelectric system under investigation with a three‐well local potential for the order parameter fluctuations. The BEG model modified by the random‐field phase diagram in the vicinity of the tricritical point qualitatively explains the influence of the cationic substitution on the (PbSn)PS ferroelectric properties.

Random Blume–Emery–Griffiths model on the Bethe lattice

The random phase transitions of the Blume–Emery–Griffiths (BEG) model for the spin-1 system are investigated on the Bethe lattice and the phase diagrams of the model are obtained. The biquadratic exchange interaction (K) is turned on, i.e. the BEG model, with probability p either attractively (K>0) or repulsively (K<0) and turned off, which leads to the BC model, with the probability (1−p) throughout the Bethe lattice. By taking the bilinear exchange interaction parameter J as a scaling parameter, the effects of the competitions between the reduced crystal fields (D/J), reduced biquadratic exchange interaction parameter (K/J) and the reduced temperature (kT/J) for given values of the probability when the coordination number is q=4, i.e. on a square lattice, are studied in detail.

Bethe lattice approach and relaxation dynamics study of spin-crossover materials

Dynamical properties of Prussian blue analogs and spin-crossover materials are investigated in the framework of a Blume–Emery–Griffiths (BEG) spin-1 model, where states ±1 and 0 represent the high-spin (HS) state and the low-spin state, respectively. The quadrupolar interaction depends on the temperature in the form $$K=\alpha k_BT$$ . Magnetic interactions are controlled by a factor $$\gamma =J/K$$ such that for $$\gamma =0$$ ( $$J=0$$ ), magnetic ordering is not expected. The model is exactly solved using the Bethe lattice approach for the equilibrium properties. The results are closer to those calculated by numerical simulations with suitable Arrhenius-type transition rates. The study of relaxation processes of non-equilibrium HS states revealed one-step nonlinear sigmoidal relaxation curves of the HS fraction at low temperatures. We found that increasing the magnetic interactions leads to the appearance of a plateau in the thermal hysteresis as well as in the relaxation curves of the HS fraction at low temperature.

Mixed Spin-2 and Spin-3/2 Blume-Emery-Griffiths (BEG) Model on the Bethe Lattice

The mixed spin-2 and spin-3/2 Blume-Emery-Griffiths (BEG) Ising ferrimagnetic system is studied by the Bethe lattice approach. The ground-state phase diagram is constructed. The influence of the crystal-field and the biquadratic interactions among neighboring spins on the thermal behaviors of the system is singled out. The system displays very rich critical behaviors with the existence of tricritical points. Compensation points where the global magnetization of the system vanishes have been detected for appropriate values of the system parameters.

Dynamic phase transitions of the Blume–Emery–Griffiths model under an oscillating external magnetic field by the path probability method

Abstract By using the path probability method (PPM) with point distribution, we study the dynamic phase transitions (DPTs) in the Blume–Emery–Griffiths (BEG) model under an oscillating external magnetic field. The phases in the model are obtained by solving the dynamic equations for the average order parameters and a disordered phase, ordered phase and four mixed phases are found. We also investigate the thermal behavior of the dynamic order parameters to analyze the nature dynamic transitions as well as to obtain the DPT temperatures. The dynamic phase diagrams are presented in three different planes in which exhibit the dynamic tricritical point, double critical end point, critical end point, quadrupole point, triple point as well as the reentrant behavior, strongly depending on the values of the system parameters. We compare and discuss the dynamic phase diagrams with dynamic phase diagrams that were obtained within the Glauber-type stochastic dynamics based on the mean-field theory.