Three-dimensional Heisenberg Ferromagnet Research Articles

Correction to scaling critical exponents and amplitudes for amorphous Fe–Mn–Zr alloys

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.

Critical magnetic properties of disordered Cr-rich FeCr alloys

A detailed analysis of high-resolution magnetisation data reveal that the critical behaviour of chemically well-homogenised high-purity disordered Cr 70Fe 30 and Cr 75Fe 25 alloys near the paramagnetic (PM)–ferromagnetic (FM) phase transition is similar to that of a three-dimensional Heisenberg ferromagnet. An ion-beam sputtered Cr 70Fe 30 thin film of 9.4 nm thickness exhibits the same critical behaviour as that of the bulk Cr 70Fe 30.

Critical behavior of electrical resistivity in amorphous Fe-Zr alloys

Electrical resistivity (ρ) of the amorphous (a-)Fe100−c Zr c (c=8.5, 9.5 and 10) alloys has been measured in the temperature range 77 to 300 K, which embraces the second-order magnetic phase transition at the Curie temperature point T c. Analysis of the resistivity data particularly in the critical region reveals that these systems have a much wider range of critical region compared to other crystalline ferromagnetic materials. The value of T c and specific heat critical exponent, α has the same values as those determined from our earlier magnetic measurements. The value of α for all the present investigated alloys are in close agreement with the values predicted for three-dimensional (3D) Heisenberg ferromagnet systems, which gives contradiction to the earlier results on similar alloys. It is observed from the analysis that the presence of quenched disorder does not have any influence on critical behavior.

Finite-Volume Excitations of the¶111 Interface in the Quantum XXZ Model

We show that the ground states of the three-dimensional XXZ Heisenberg ferromagnet with a 111 interface have excitations localized in a subvolume of linear size R with energies bounded by O(1/R^2). As part of the proof we show the equivalence of ensembles for the 111 interface states in the following sense: In the thermodynamic limit the states with fixed magnetization yield the same expectation values for gauge invariant local observables as a suitable grand canonical state with fluctuating magnetization. Here, gauge invariant means commuting with the total third component of the spin, which is a conserved quantity of the Hamiltonian. As a corollary of equivalence of ensembles we also prove the convergence of the thermodynamic limit of sequences of canonical states (i.e., with fixed magnetization).

Critical phenomena of amorphous Nd 4(Fe 0.75Cr 0.25) 77.5B 18.5 alloys

The magnetic behavior of amorphous Nd 4(Fe 0.75Cr 0.25) 77.5B 18.5 alloys was investigated in the critical region. The Curie temperature T C and critical exponents β, γ and δ are found to be 141 K, 0.45±0.02, 1.64±0.08 and 4.66±0.10, respectively. The data are fitted to a magnetic equation of state characteristic of a second-order phase transition over a rather wide range of temperatures both above and below T C. It is noted that the values of the exponents are in disagreement with those derived for a three-dimensional Heisenberg ferromagnet and show an enhancement. This anomalous critical behavior may originate from magnetic inhomogeneity.

Zigzag-chain structure and ferromagnetism of

The crystal structure of the insulator has been determined at 296 K by means of x-ray four-circle diffraction with residual factors of R = 0.027 and . The crystal data show an orthorhombic structure; space group Pnma, a = 14.301(1) Å, b = 3.598(1) Å, c = 5.204(1) Å, and Z = 4. The isolated V zigzag chains with a nearest-neighbour V-V distance of 3.05 Å are formed along the b-axis by sharing edges and corners of pyramids. The zigzag-chain unit and the pyramid are almost identical to those of the two-dimensional system which exhibits a simple dimer state with a gap of 660 K. The electronic states have been explored through magnetization and electron paramagnetic resonance measurements. is an ferromagnet with a Curie temperature . The paramagnetic properties above 50 K are explained in terms of the ferromagnetic Heisenberg chain model in the classical limit, where g = 1.96 and the edge-sharing exchange coupling , and the properties between 50 K and 30 K are considered as indicating a critical behaviour of the three-dimensional Heisenberg ferromagnet. This evidence is consistent with a previous model for the spin-gap formation of , in which it was postulated that the exchange coupling between the zigzag chains is rather larger than those within the zigzag chain.

Dynamic critical exponents and sample independence times for the classical Heisenberg model.

The critical dynamics of the scalar order parameter autocorrelation function, the total vector spin autocorrelation function, and the energy autocorrelation function for the classical O(3) three-dimensional Heisenberg ferromagnet are studied both algebraically and with heat bath Monte Carlo simulations. It is shown from a group decomposition of the canonical Markov transitions used in a Monte Carlo study of this model ferromagnet that the vector spin autocorrelation relaxation time currently used to estimate both the dynamical critical exponent and the statistical sampling efficiency of the scalar order parameter (and other quantities of interest) contains an uncontrolled dependence upon irrelevant symmetric diffusion processes and is therefore unsuitable for this purpose. The central limit theorem is quantified as the fundamental basis of estimates of error and, separately, estimates of statistical sampling efficiency in importance sampling Monte Carlo simulations. A simple but interesting estimate of ${\mathit{z}}_{\mathit{R}}$=d-2\ensuremath{\beta}/\ensuremath{\nu} for the dynamical critical exponent of the ergodicity-restoring thermal-rotational motion of the total vector spin is presented. Various relaxation times are numerically measured and their dynamic critical exponents extracted via finite-size scaling theory to illustrate these points and compare with theoretical predictions. \textcopyright{} 1996 The American Physical Society.

The magnetic phase transition in Fe 17Dy 2 and Fe 23Er 6 single crystals

The magnetic phase transition in Fe 17Dy 2 and Fe 23Er 6 single crystals is investigated from detailed magnetisation measurements performed in the critical region. Two sets of measurements were performed while keeping the direction of the applied magnetic field along the easy plane as well as hard directions. Values of the Curie temperature ( T C) and the critical exponents corresponding to the spontaneous magnetisation (β), initial susceptibility (γ′, γ) and the magnetisation isotherm at the Curie point (δ) are deduced from the analysis based on several methods: modified Arrott plots, the Kouvel-Fisher method, scaling plots and ln J versus ln( μ 0 H) plots. Very good agreement has been observed between the exponent values obtained by the various methods. The values of β, γ and δ for Fe 17Dy 2 are close to those predicted theoretically for a three-dimensional Heisenberg ferromagnet, while those for Fe 23Er 6 do not belong to any known universality class. The exponents deduced from the measurements in the easy plane and hard directions vary only within our error limits.

Determination of asymptotic critical exponents and amplitudes for amorphous 3D ferromagnets from bulk magnetization measurements

High-resolution bulk magnetization data have been taken in the critical region (−0.05 ≤ ϵ = ( T − T C)/ T C ≤ 0.05, where T C is the Curie temperature) on amorphous Fe x Ni 80- x P 14B 6 alloys with x = 20, 30 and 40. An elaborate data analysis, besides yielding accurate values for the spontaneous magnetization and initial susceptibility asymptotic (leading ‘correction-to-scaling’) critical exponents and amplitudes (amplitudes), reveals that the values of the asymptotic critical exponents and universal amplitude ratios m o/ M S( O) and Dm o δ / h o are composition-independent and exactly equal to those for the pure (ordered) spin system with space as well as spin dimensionality of three. However, the main difference between crystalline (ordered) and amorphous (quench-disordered) three-dimensional Heisenberg ferromagnets is that nearly all the spins participate in the ferromagnetic (FM)-paramagnetic (PM) phase transition in the former case while only a small fraction of spins is actually involved in such a transition in the latter case. This fraction reduces drastically as the percolation threshold for the long-range FM order is approached along the FMPM phase transition line in the magnetic phase diagram. Asymptotic critical exponents β, γ and δ obey Widom scaling relation, βδ = β + γ, and the magnetization data satisfy the scaling equation of state for second-order phase transition in the asymptotic critical region.

Detailed magnetization study of quenched random ferromagnets. II. Site dilution and percolation exponents.

We report the experimental determination of the crossover exponent (\ensuremath{\varphi}) and the percolation critical exponents for magnetization (${\mathrm{\ensuremath{\beta}}}_{\mathit{p}}$) and spin-wave stiffness (\ensuremath{\theta}) for quench-disordered (amorphous) three-dimensional (d=3) dilute Heisenberg ferromagnets. The values of \ensuremath{\varphi}, \ensuremath{\theta}, and ${\mathrm{\ensuremath{\beta}}}_{\mathit{p}}$, so obtained, as well as those of the percolation correlation-length critical exponent ${\ensuremath{\nu}}_{\mathit{p}}$ and the conductivity exponent \ensuremath{\sigma}, deduced from the exponent equalities ${\ensuremath{\nu}}_{\mathit{p}}$=\ensuremath{\varphi}-(\ensuremath{\theta}/2) and \ensuremath{\sigma}=(d-2)${\ensuremath{\nu}}_{\mathit{p}}$+\ensuremath{\varphi}, conform very well with the most accurate theoretical estimates published recently. A comparison of the presently determined exponent values with those theoretically predicted for site or bond percolation on a d=3 crystalline lattice asserts that the critical behavior of percolation on a regular d=3 lattice does not get altered in the presence of quenched randomness if the specific-heat exponent of the regular system is negative. Consistent with the Alexander-Orbach conjecture (Golden inquality), the fracton dimensionality d\ifmmode\bar\else\textasciimacron\fi{}\ifmmode\bar\else\textasciimacron\fi{} of the percolating cluster at threshold (the conductivity exponent \ensuremath{\sigma}) turns out to be d\ifmmode\bar\else\textasciimacron\fi{}\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\simeq}4/3 (\ensuremath{\sigma}\ensuremath{\le}2). The present results vindicate the universality hypothesis.

Spin-dynamics study of the dynamic critical behavior of the three-dimensional classical Heisenberg ferromagnet.

Using spin-dynamics techniques we have performed large-scale computer simulations of the dynamic behavior of the L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L body-centered-cubic classical Heisenberg ferromagnet with L\ensuremath{\le}40 in the vicinity of the critical point ${\mathit{T}}_{\mathit{c}}$. The temporal evolutions of spin configurations were determined numerically from coupled equations of motion for individual spins by a fourth-order predictor-corrector method with initial spin configurations generated by Monte Carlo methods. The space- and time-displaced spin-spin correlation functions and their space-time Fourier transforms were calculated to determine the neutron-scattering functions. We developed a dynamic finite-size scaling theory for the neutron-scattering function at ${\mathit{T}}_{\mathit{c}}$ and used it to extract the dynamic critical exponent z. Within our resolution limit, the value of z was estimated to be 2.478(28), in excellent agreement with the dynamic scaling prediction, and dynamic scaling was found to be valid for momentum transfer q up to 0.4\ensuremath{\pi} in the (1,0,0) direction.

Monte Carlo study of critical behavior in the three-dimensional classical Heisenberg ferromagnet

The static critical properties of classical, isotropic Heisenberg ferromagnets have been studied numerically. Using an improved cluster algorithm we performed extensive Monte Carlo simulations on L×L×L simple cubic and body-centered cubic systems with L≤40 over a temperature range covering the critical region. Temperature dependences of thermodynamic quantities were determined with an optimized multiple-histogram method, and both critical temperatures and critical exponents were extracted from finite-size scaling of several different thermodynamic quantities. It was found that within their respective error bars the critical exponents for both ferromagnets agree with each other and with field theoretical predictions.

Magnetic percolation threshold and critical exponents of Ni80_xFexB18Si2 amorphous alloys

Abstract The magnetic and electronic transport properties of Ni80-xFexB18Si2, amorphous alloys, with x being in the vicinity of the ferromagnetic percolation threshold xc, have been investigated. The results indicate that xc, is close to x = 2.3 and yield well defined Curie temperatures Tc for all alloys with x ⩾ xc. In spite of this, the critical exponent of the magnetic susceptibility γ can be determined only for the alloys with x ⩾ 2.6. The value of γ for these alloys is independent of x and is close to that calculated for a three-dimensional Heisenberg ferromagnet.

High-accuracy Monte Carlo study of the three-dimensional classical Heisenberg ferromagnet.

Using extensive Monte Carlo simulations, we study the equilibrium properties of the simple-cubic, classical Heisenberg ferromagnet. We employ very long runs for L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L lattices to obtain high-precision data for the magnetization probability distribution. Using finite-size scaling for L\ensuremath{\le}24 and an optimized multiple-histogram data analysis, we obtain an accurate value of the inverse critical temperature J/${\mathit{k}}_{\mathit{B}}$${\mathit{T}}_{\mathit{c}}$=0.6929\ifmmode\pm\else\textpm\fi{}0.0001, which is higher than previously accepted estimates. Calculated values of various static exponents are in excellent agreement with renormalization-group and \ensuremath{\epsilon}-expansion predictions.

Extension of the random-phase-approximation theory of ferromagnetism in a magnetic field applicable at all temperatures.

Generalization of the random-phase-approximation (RPA) theory of ferromagnetism for spins S=1/2, based on a refinement of the decoupling for a relevant three-spin Green's function, is presented. With the reasonable choice of two characteristic functions of the present theory---the virtual spectrum \ensuremath{\Delta}(\ensuremath{\omega}) and the decoupling factor \ensuremath{\kappa}(\ensuremath{\sigma}), where \ensuremath{\omega} is an excitation energy and \ensuremath{\sigma} is the relative magnetization---the theory provides a fair description of magnetic phenomena not only at temperature T\ensuremath{\rightarrow}0, in the vicinity of the Curie temperature ${\mathit{T}}_{\mathit{C}}$, or at high T, but at all temperatures and external magnetic fields H. A relation between some parameters of the functions \ensuremath{\Delta}(\ensuremath{\omega}) and \ensuremath{\kappa}(\ensuremath{\sigma}), and the critical indices of the static scaling theory, has been established. The form of the universal scaling function W, following from our theory, has been found. For the power-law virtual spectrum \ensuremath{\Delta}\ensuremath{\propto}${\mathrm{\ensuremath{\omega}}}^{\mathit{s}}$ at \ensuremath{\omega}\ensuremath{\rightarrow}0, 0s1, we have found the nonanalytic increase of the magnetization versus the field at T${\mathit{T}}_{\mathit{C}}$:\ensuremath{\sigma}(T,H)-\ensuremath{\sigma}(T,0)\ensuremath{\propto}${\mathit{H}}^{\mathit{s}}$. It reduces to the classic result of Holstein and Primakoff for the three-dimensional isotropic Heisenberg ferromagnet having s=1/2. Assuming this power-law spectrum at all energies up to some cutoff energy, we have obtained a rather good agreement of the present predictions for \ensuremath{\sigma}(T,H) with experimental data for Ni, for all temperatures and fields available. Some model virtual spectra are found to lead to interesting models of ferromagnetism, with the magnetization and the susceptibility functions given in terms of explicit analytic formulas.

Spin-wave localization in ferromagnets with random anisotropy

We argue that in a three-dimensional Heisenberg ferromagnet with weak random-axes anisotropy and large magnetization, induced by the magnetic field, the low-frequency spin waves will be localized. Two regimes are distinguished: in smaller fields, the regime with the field-dependent correlations of the transverse magnetization, i.e., the ferromagnet with wandering axis (FWA); in larger fields, the mean-field (MF) regime with the transverse magnetization uncorrelated from site to site. Using the approach developed for ferromagnets with uniaxial random-strength anisotropy, we find that the mobility edge in both FWA and MF regimes scales as λmob∝(ΔM/M)−1. Here λ is the spin wavelength and ΔM is the deviation from the saturation magnetization M. It is known that ΔM∝H−1/2 in the FWA regime and ΔM∝H−2 in the MF regime. Thus, the mobility edge can be tuned by the magnetic field.

Asymptotic and leading "correction-to-scaling" susceptibility critical exponents and amplitudes for ferromagnets with quenched disorder.

Accurate values of the asymptotic susceptibility critical exponent (\ensuremath{\gamma}), critical amplitude (\ensuremath{\Gamma}), and the leading ``correction-to-scaling'' exponents and amplitudes for quench-disordered ferromagnets have been determined for the first time through an elaborate analysis of precise and comprehensive ac susceptibility data taken on amorphous ${\mathrm{Fe}}_{\mathrm{x}}$${\mathrm{Ni}}_{80\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{B}}_{19}$${\mathrm{Si}}_{1}$(x =10, 13, and 16) alloys in the critical region. Consequently, the present work provides unambiguous experimental verification of theoretical predictions based on renormalization-group (RG) calculations, and testifies to their correctness by demonstrating that the exponent \ensuremath{\gamma} of the ordered three-dimensional Heisenberg ferromagnet is not affected by the presence of quenched disorder and that the ``frozen'' disorder plays the role of an irrelevant scaling field (in the RG sense).

Monte Carlo calculation of free energy, critical point, and surface critical behavior of three-dimensional Heisenberg ferromagnets.

A transfer-matrix Monte Carlo technique is developed to compute the free energy of three-dimensional, classical Heisenberg ferromagnets. From the free energy of systems with periodic and antiperiodic boundary conditions, helicity moduli are calculated. From these the critical couplings for simple-cubic (sc) and face-centered-cubic lattices are estimated by use of finite-size scaling. For the simple-cubic lattice, the critical dimension of the surface magnetization is estimated with standard Monte Carlo methods, yielding a result in excellent agreement with the \ensuremath{\epsilon}-expansion work of Diehl and N\usser.

Three-dimensional Heisenberg ferromagnet: A series investigation.

We present new twelfth-order high-temperature series for the susceptibility, correlation length, and free energy of the nearest-neighbor classical Heisenberg ferromagnet in zero field on a fcc lattice. Checks corroborating the correctness of the series are discussed. Standard analysis of these series produces results in essential agreement with the renormalization-group calculations and several other series analyses. However, a more reliable confluent singularity analysis suggests slightly larger values, \ensuremath{\gamma}${=1.40}_{\mathrm{\ensuremath{-}}0.01}^{+0.03}$ and \ensuremath{\nu}=0.72\ifmmode\pm\else\textpm\fi{}0.01, as well as a confluent correction of just the size to cause the observed standard analysis underestimate.

The magnetic phase transition in amorphous ferromagnets and in spin glasses

The magnetic phase transition in materials with exchange disorder (amorphous ferromagnets, spin glasses) is discussed. In the critical temperature range the behavior of amorphous ferromagnetic transition metal-metalloid glasses is found to be similar to the one derived for a three-dimensional homogeneous Heisenberg ferromagnet. The most prominent difference between disordered and homogeneous materials is manifested in a large temperature range of deviations from the mean field behavior beyond the critical region, as observed experimentally for the temperature dependence of the linear susceptibility of amorphous ferromagnets and of the nonlinear susceptibility of spin glasses. A molecular field theory with correlations in space and time is developed, which relates the deviations from the mean field behavior to the interplay between the temperature dependent thermal correlations in the spin system and the spatial fluctuations of the material. Application to dynamical processes (kinetic critical slowing down) is discussed.