Three-dimensional Ising Ferromagnet Research Articles

Different universality classes of isostructural UTX compounds (T=Rh,Co,Co0.98Ru0.02;X=Ga,Al)

Magnetization isotherms of the $5f$-electron ferromagnets URhGa, UCoGa, and $\mathrm{U}{\mathrm{Co}}_{0.98}{\mathrm{Ru}}_{0.02}\mathrm{Al}$ were measured at temperatures in the vicinity of their Curie temperature to investigate the critical behavior near the ferromagnetic phase transition. These compounds adopt the layered hexagonal ZrNiAl-type structure and exhibit huge uniaxial magnetocrystalline anisotropy. The critical \ensuremath{\beta}, \ensuremath{\gamma}, and \ensuremath{\delta} exponents were determined by analyzing Arrott-Noakes plots, Kouvel-Fisher plots, critical isotherms, scaling theory, and Widom scaling relations. The values obtained for URhGa and UCoGa can be explained by the results of the renormalization group theory for a two-dimensional (2D) Ising system with long-range interactions similar to URhAl reported by other investigators. On the other hand, the critical exponents determined for $\mathrm{U}{\mathrm{Co}}_{0.98}{\mathrm{Ru}}_{0.02}\mathrm{Al}$ are characteristic of a three-dimensional (3D) Ising ferromagnet with short-range interactions suggested in previous studies also for the itinerant $5f$-electron paramagnet UCoAl situated near a ferromagnetic transition. The change from the 2D to the 3D Ising system is related to the gradual delocalization of $5f$ electrons in the series of the URhGa, URhAl, and UCoGa to $\mathrm{U}{\mathrm{Co}}_{0.98}{\mathrm{Ru}}_{0.02}\mathrm{Al}$ and UCoAl compounds and appears close to the strongly itinerant nonmagnetic limit. This indicates possible new phenomena that may be induced by the change of dimensionality in the vicinity of the quantum critical point.

Dynamic phase transition of the Blume-Capel model in an oscillating magnetic field.

We employ numerical simulations and finite-size scaling techniques to investigate the properties of the dynamic phase transition that is encountered in the Blume-Capel model subjected to a periodically oscillating magnetic field. We mainly focus on the study of the two-dimensional system for various values of the crystal-field coupling in the second-order transition regime. Our results indicate that the present nonequilibrium phase transition belongs to the universality class of the equilibrium Ising model and allow us to construct a dynamic phase diagram, in analogy with the equilibrium case, at least for the range of parameters considered. Finally, we present some complementary results for the three-dimensional model, where again the obtained estimates for the critical exponents fall into the universality class of the corresponding three-dimensional equilibrium Ising ferromagnet.

Zero-temperature relaxation of three-dimensional Ising ferromagnets

We investigate the properties of the Ising-Glauber model on a periodic cubic lattice of linear dimension L after a quench to zero temperature. The resulting evolution is extremely slow, with long periods of wandering on constant energy plateaus, punctuated by occasional energy-decreasing spin-flip events. The characteristic time scale τ for this relaxation grows exponentially with the system size; we provide a heuristic and numerical evidence that τ~exp(L(2)). For all but the smallest-size systems, the long-time state is almost never static. Instead, the system contains a small number of "blinker" spins that continue to flip forever with no energy cost. Thus, the system wanders ad infinitum on a connected set of equal-energy blinker states. These states are composed of two topologically complex interwoven domains of opposite phases. The average genus g(L) of the domains scales as L(γ), with γ≈1.7; thus, domains typically have many holes, leading to a "plumber's nightmare" geometry.

Nonequilibrium magnetization reversal by periodic impulsive fields in Ising mean-field dynamics

We studied nonequilibrium magnetization reversal in kinetic Ising ferromagnets driven by a periodic impulsive magnetic field in the mean-field approximation. The mean-field differential equation was solved by the sixth-order Runge–Kutta–Felberg method. The periodicity and strength of the applied impulsive magnetic field play a key role in magnetization reversal. We studied the minimum strength of an impulsive field required for magnetization reversal at any temperature as a function of the periodicity of the impulsive field. At high temperature and small period, this was observed to be linear. The results were compared with those obtained from the Monte Carlo simulation of a three-dimensional Ising ferromagnet.

Extended scaling for ferromagnets

A simple systematic rule, inspired by high-temperature series expansion (HTSE) results, is proposed for optimizing the expression for thermodynamic observables of ferromagnets exhibiting critical behavior at ${T}_{c}$. This ``extended scaling'' scheme leads to a protocol for the choice of scaling variables, $\ensuremath{\tau}=(T\ensuremath{-}{T}_{c})∕T$ or $({T}^{2}\ensuremath{-}{T}_{c}^{2})∕{T}^{2}$ depending on the observable instead of $(T\ensuremath{-}{T}_{c})∕{T}_{c}$, and more importantly to temperature dependent noncritical prefactors for each observable. The rule corresponds to scaling of the leading term of the reduced susceptibility above ${T}_{c}$ as ${\ensuremath{\chi}}_{c}^{*}(T)\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}\ensuremath{\gamma}}$ in agreement with standard practice with scaling variable $\ensuremath{\tau}$ and for the leading term of the second-moment correlation length as ${\ensuremath{\xi}}_{c}^{*}(T)\ensuremath{\sim}{T}^{\ensuremath{-}1∕2}{\ensuremath{\tau}}^{\ensuremath{-}\ensuremath{\nu}}$. For the specific heat in bipartite lattices, the rule gives ${C}_{c}^{*}(T)\ensuremath{\sim}{T}^{\ensuremath{-}2}{[({T}^{2}\ensuremath{-}{T}_{c}^{2})∕{T}^{2}]}^{\ensuremath{-}\ensuremath{\alpha}}$. The latter two expressions are not standard. The scheme can allow for confluent and noncritical correction terms. A stringent test of the extended scaling is made through analyses of high-precision numerical and HTSE data, or real data, on the three-dimensional canonical Ising, $XY$, and Heisenberg ferromagnets. For the susceptibility $\ensuremath{\chi}(T)$ and the correlation length $\ensuremath{\xi}(T)$ of the three ferromagnets, their optimized expression, which consists of the leading terms [respectively, ${\ensuremath{\chi}}_{c}^{*}(T)$ and ${\ensuremath{\xi}}_{c}^{*}(T)$] and a quite limited number of confluent and noncritical correction terms, represents real data to surprisingly good approximations over the entire temperature range from ${T}_{c}$ to infinity. The temperature dependent prefactors introduced are of crucial importance not only in fixing the optimized expression at relatively high temperatures but also in determining appropriately the small amplitude correction terms. For the specific heat of the Ising ferromagnet, ${C}_{c}^{*}(T)$ combined with two noncritical correction terms which are calculated with no free parameters once the correlation length critical parameters are known reproduces real data nicely also over the whole temperature range.

Overlap distribution of the three-dimensional Ising model.

We study the Parisi overlap probability density P(L)(q) for the three-dimensional Ising ferromagnet by means of Monte Carlo (MC) simulations. At the critical point, P(L)(q) is peaked around q=0 in contrast with the double peaked magnetic probability density. We give particular attention to the tails of the overlap distribution at the critical point, which we control over up to 500 orders of magnitude by using the multioverlap MC algorithm. Below the critical temperature, interface tension estimates from the overlap probability density are given and their approach to the infinite volume limit appears to be smoother than for estimates from the magnetization.

Thermodynamics of three-dimensional Ising-like systems in the higher non-Gaussian approximation: Calculational method and dependence on microscopic parameters

The original collective variables method for calculating the thermodynamic characteristics of a three-dimensional Ising-like system is developed in the approximation of the sextic distribution for modes of spin-density oscillations (ρ 6 model), taking into account the first confluent correction. The calculations within the framework of this higher non-Gaussian approximation are performed in the temperature regions above and below the critical value of T c and are illustrated by an example of a simple cubic lattice and an exponentially decreasing interaction potential. The collective variables method makes it possible to obtain universal and nonuniversal quantities by using a unified approach. The dependence of the critical region size and phase-transition temperature on the microscopic parameters is investigated. A microscopic analog of the Landau free energy is calculated. Explicit expressions for the thermodynamic characteristics are obtained as functions of temperature and microscopic parameters of the system. It is shown that each of the leading critical amplitudes and correction-to-scaling amplitudes can be represented as a product of a universal factor not depending on the microscopic parameters of the system and a nonuniversal factor depending on these parameters. The thermodynamic characteristics near T c and their amplitudes are given for various values of the effective radius of the exponentially decreasing interaction potential (including the values corresponding to the nearest-neighbor interaction and the interactions between the nearest and next-nearest neighbors and between the nearest, next-nearest, and third neighbors). The results of calculations and their comparison with other authors' data show that the ρ 6 model provides a better quantitative description of the critical behavior of a three-dimensional Ising ferromagnet than the ρ 4 model.

`Fragile' glass-forming behaviour ofa three-dimensional Ising ferromagnet with uniformfrustration

By using Monte Carlo simulation, we study the dynamics around equilibrium of a uniform frustrated three-dimensionalIsing model with nearest-neighbour ferromagnetic interactions and small, but long-range (coulombic) antiferromagnetic interactions. This model exhibits a strong slowing down of the relaxations for temperatures below the critical temperature of the unfrustrated model. The characteristics ofthis slowing down - super-Arrhenius activated temperaturedependence of the relaxation time and non-exponential decayof the correlation function - are similar to those observed in fragile glass-forming liquids.

Computer simulations of a three-dimensional Ising ferromagnet with quenched disorder

Recent high-resolution x-ray and neutron-scattering experiments for various crystals with quenched defects of the random temperature type near magnetic and structural phase transitions have revealed two different length scales. The coexistence of two scales was interpreted as a superposition effect of scattering from the disordered layer containing defects, i.e., the surface layer (resulting in the larger correlation length) and from the bulk of the sample (resulting in the shorter one which corresponds to the usual critical fluctuation). Later, experimental evidence was found for the coexistence of both scales in the same volume fraction of the sample. In our investigations the phase transition in an Ising ferromagnet containing edge dislocation dipoles is considered. The results of our computer simulations also show that the two different length scales observed in real crystals appear in the same volume fraction.

Critical behaviour of three-dimensional Ising ferromagnets at imperfect surfaces: Bounds on the surface critical exponent

The critical behaviour of three-dimensional semi-infinite Ising ferromagnets at planar surfaces with (i) random surface-bond disorder or (ii) a terrace of monatomic height and macroscopic size is considered. The Griffiths-Kelly-Sherman correlation inequalities are shown to impose constraints on the order-parameter density at the surface, which yield upper and lower bounds for the surface critical exponent β 1. If the surface bonds do not exceed the threshold for supercritical enhancement of the pure system, these bounds force β 1 to take the value β 1 ord of the latter system's ordinary transition. This explains the robustness of β 1 ord to such surface imperfections observed in recent Monte Carlo simulations.

Modified scaling relation for the random-field Ising model

We investigate the low-temperature critical behavior of the three-dimensional random-field Ising ferromagnet. By a scaling analysis we find that in the limit of temperature T→0 the usual scaling relations have to be modified as far as the exponent α of the specific heat is concerned. At zero temperature, the Rushbrooke equation is modified to α+2 β+ γ=1, an equation which we expect to be valid also for other systems with similar critical behavior. We test the scaling theory numerically for the three-dimensional random-field Ising system with Gaussian probability distribution of the random fields by a combination of calculations of exact ground states with an integer optimization algorithm and Monte Carlo methods. By a finite-size scaling analysis we calculate the critical exponents ν≈1.0, β≈0.05, γ ̄ ≈2.9 , γ≈1.5 and α≈−0.55.

Universal effective action for the three-dimensional ising ferromagnet

We compute the universal effective action for one-component real scalar field in three Euclidean dimensions, in the case of spontaneously broken symmetry, from the Monte Carlo simulation of the three-dimensional Ising model. We find that, in close analogy with the symmetric case, the effective potential is very well approximated by the sum of φ 2 , φ 4 and φ 6 terms. An unexpected feature is the negative sign of the φ 4 term.

Curie temperature for three-dimensional binary Ising ferromagnets.

Using the Swendsen and Wang algorithm, high accuracy Monte Carlo simulations were performed to study the concentration dependence of the Curie temperature in binary, ferromagnetic Ising systems on the simple-cubic lattice. Our results are in good agreement with known mean-field like approaches. Based on former theoretical formulas we propose a new way of estimating the Curie temperature of these systems.

“ENVIRONMENTALLY FRIENDLY” RENORMALIZATION

We analyze the renormalization of systems whose effective degrees of freedom are described in terms of fluctuations which are “environment”-dependent. Relevant environmental parameters considered are: temperature, system size, boundary conditions and external fields. The points in the space of “coupling constants” at which such systems exhibit scale invariance coincide only with the fixed points of a global renormalization group which is necessarily environment-dependent. Using such a renormalization group we give formal expressions to two loops for effective critical exponents for a generic crossover induced by a relevant mass scale g. These effective exponents are seen to obey scaling laws across the entire crossover, including hyperscaling, but in terms of an effective dimensionality, deff=4−γλ, which represents the effects of the leading irrelevant operator. We analyze the crossover of an O(N) model on a d-dimensional layered geometry with periodic, antiperiodic and Dirichlet boundary conditions. Explicit results to two loops for effective exponents are obtained using a [2/1] Padé-resummed coupling, for: the “Gaussian model” (N=−2), spherical model (N=∞), Ising model (N=1), polymers (N=0), XY model (N=2) and Heisenberg (N=3) models in four dimensions. We also give two-loop Padé resummed results for a three-dimensional Ising ferromagnet in a transverse magnetic field and corresponding one-loop results for the two-dimensional model. One-loop results are also presented for a three-dimensional layered Ising model with periodic, Dirichlet and antiperiodic boundary conditions. Asymptotically the effective exponents are in excellent agreement with known results.

Erratum: Generalized difference equation for the three-dimensional Ising ferromagnet

Erratum: Generalized difference equation for the three-dimensional Ising ferromagnet

Generalized difference equation for the three-dimensional Ising ferromagnet

The difference equation method of Frank, Cheung, and Mouritsen (FCM) is made self-consistent by relaxing an assumption of the theory which pertains to the correlation of a spin with its nearest neighbour. The full set of A = 1 and I − δ equations and U equations is brought into play, with the U2 = T2 = (1/4) (1 − 1/F(1)) double equation replaced by a single relation between U2 and T2. The resulting ferromagnetic transition temperatures for the cubic lattices are consistent with the series results to within about 2%. The correlation functions are much closer to the series results than those of FCM.

Abstract: Critical behavior in ferromagnetic Fe[S2CN(C2H5)2]2Cl

Experimental work on the three-dimensional Ising ferromagnet Fe[S2CN(C2H5)2]2Cl is reviewed, and magnetic susceptibility and sublattice magnetization2 data are analyzed with correction-to-scaling terms included. Critical exponents γ = 1.165±0.03 and β = 0.245±0.02 are found, assuming modified power law expressions χ0 = Γt−γ (1+aχtΔ) and m(T/TC) = B‖t‖β(1+aM ‖t‖Δ) with correction exponent and amplitudes, Δ = 0.493, aχ = 0.207, and aM = 0.156. The critical amplitudes are found to be Γ = 1.47±0.07 emu/mole (Γr = 1.07±0.05 for the reduced susceptibility) and B = 1.30±0.05. Γ is larger than most theoretical estimates for the Ising model on three-dimensional lattices, while B is smaller than theoretical results. The amplitude ratio Γr/B is 0.83±0.07. Theoretical values for this ratio are not universal but exhibit only a weak lattice dependence, and are in the range from 0.65 to 0.71 for the three-dimensional Ising model. Heat capacity data are sparse and less reliably analyzed, but appear to suggest abnormally large values of α (≳0.3) and rather small values of A (0.2–0.5) in C/R = At−α+ B′. Including a correction-to-scaling term does not alter this conclusion. The ratio RC = AB−2Γr is in the range from 0.12 to 0.33, substantially less than the theoretical value 0.474 for this universal quantity in the three-dimensional Ising model. Neither the possible presence of multispin interactions nor a nondiverging AF contribution to the susceptibility appear to be adequate to account for the unusual critical behavior observed in this system. a)Acknowledgment is made to the donors of The Petroleum Research Fund, administered by the ACS, for partial support of this research. 1 G. C. DeFotis, F. Palacio, and R. L. Carlin, Phys. Status Solidi B 95, 380 (l978). 2 G. C. DeFotis and J. A. Cowen, J. Chem. Phys. 73, 2120 (1980).

Random-Field Effects in Two- and Three-Dimensional Ising Antiferromagnets

Neutron scattering experiments have been performed in the diluted antiferromagnets ${\mathrm{Rb}}_{2}$${\mathrm{Co}}_{0.7}$${\mathrm{Mg}}_{0.3}$${\mathrm{F}}_{4}$ and ${\mathrm{Co}}_{0.3}$${\mathrm{Zn}}_{0.7}$${\mathrm{F}}_{2}$ in a uniform magnetic field. These systems are isomorphous to, respectively, two- and three-dimensional Ising ferromagnets in a site-random magnetic field. It is shown that small random magnetic fields destroy the long-range order at all temperatures, consistent with three as the lower marginal dimensionality; the structure factor in the disordered state is predominantly a squared Lorentzian.

Renormalization group phase boundary of the three-dimensional bond-dilute Ising ferromagnet

The phase boundary (as well as the thermal-type critical exponents) associated to the quenched bond-dilute spin-1/2 Ising ferromagnet in the simple cubic lattice is approximately calculated within a real space renormalization group framework in two different versions. Both lead to qualitatively satisfactory critical frontiers, although one of them provides an unphysical fixed point (which seems to be related to the three-dimensionality of the system) besides the expected pure ones; its effects tend to disappear for increasingly large clusters. Through an extrapolation procedure the (unknown) critical phase boundary is approximately located.

Improved product average decomposition method for the Ising ferromagnet with two- and four-spin interaction

Two improvements to a previous calculation of the transition temperature for the three-dimensional Ising ferromagnet with combined two- and four-spin interaction are presented. The first improvement follows from the complete implementation of a refined product average decomposition (PAD) method which distinguishes between nearest-neighbour and non-nearest-neighbour pairs. The second improvement comes from a comparison of the value for the ordinary two-spin interaction average of order four calculated using the PAD method, with its more accurate value calculated using a critical correlation-function approximation method. The comparison with the available values of the transition temperature for the FCC lattice obtained from low-temperature expansions shows that the errors are below 0.1% at alpha = infinity ( alpha =J4/J2), below 5.8% for alpha <or=2, and below 2.4% for alpha <or=1. At alpha =0 the errors are less than 0.3%. For intermediate values of alpha (4<or= alpha <or=8, for example) the errors are about 11%.