Classical Mean Field Research Articles

Reduction of quantum fluctuations by anisotropy fields in Heisenberg ferro‐ and antiferromagnets

AbstractThe physical properties of quantum systems, which are described by the anisotropic Heisenberg model, are influenced by thermal as well as by quantum fluctuations. Such a quantum Heisenberg system can be profoundly changed towards a classical system by tuning two parameters, namely the total spin and the anisotropy field: Large easy‐axis anisotropy fields, which drive the system towards the classical Ising model, as well as large spin quantum numbers suppress the quantum fluctuations and lead to a classical limit. We elucidate the incipience of this reduction of quantum fluctuations. In order to illustrate the resulting effects we determine the critical temperatures for ferro‐ and antiferromagnets and the ground state sublattice magnetization for antiferromagnets. The outcome depends on the dimension, the spin quantum number and the anisotropy field and is studied for a widespread range of these parameters. We compare the results obtained by: Classical Mean Field, Quantum Mean Field, Linear Spin Wave and Random Phase Approximation. Our findings are confirmed and quantitatively improved by numerical Quantum Monte Carlo simulations. The differences between the ferromagnet and antiferromagnet are investigated. We finally find a comprehensive picture of the classical trends and elucidate the suppression of quantum fluctuations in anisotropic spin systems. In particular, we find that the quantum fluctuations are extraordinarily sensitive to the presence of small anisotropy fields. This sensitivity can be quantified by introducing an “anisotropy susceptibility”.

Ion binding to polyelectrolytes: Monte Carlo simulations versus classical mean field theories

The influence of ion size and surface charge model in titrations of ionizable polyelectrolytes is studied by means of the Semi Grand Canonical Monte Carlo simulation method in the context of the primitive model. Three models describing a discrete distribution of charged functional groups on the polyelectrolyte and different values for the radius of the background electrolyte spanning from ionic to hydrated radii values were analyzed. The polyelectrolyte titrations were simulated by calculating the degree of ionization versus pH curves at two ionic strengths. The results allow us to quantify the impact of the sizes of the background salt ions and surface functional groups of the polyelectrolyte on the dissociation degree. This influence is explained in terms of the effectiveness of the screening of the charged surface sites. Finally, by comparison with the Non-Linear Poisson–Boltzmann model, the influence of ionic correlations and finite size of the solution ions is assessed.

Transport properties near the magneto/structural transition of Tb5Si2Ge2

Abstract We report high resolution measurements of the electrical resistivity (ρ, dρ/dT) and thermopower (S, dS/dT) measurements near the magnetic and structural transition of the layered Tb5Si2Ge2 compound, which are fairly close but not fully coupled. The analysis of the transport properties confirms a split magneto/structural transition, with TS ∼ 97 K and TS ∼ 107 K for the structural transition (on cooling and heating respectively; 1st-order transition). The magnetic transition occurs only at TC ∼ 112 K and without hysteresis (2nd-order transition). The magnetic critical behavior of resistivity is analyzed, obtaining an almost classical mean field exponent (α ∼ 0.59) for T > TC. For the structural phase, and below TS, we obtain a rather different exponent (α ∼ 1.06).

Simplex solid states ofSU(N)quantum antiferromagnets

I define a set of wave functions for $\mathrm{SU}(N)$ lattice antiferromagnets, analogous to the valence bond solid states of Affleck et al. [Phys. Rev. Lett. 59, 799 (1987); Commun. Math. Phys. 115, 477 (1988)], in which the singlets are extended over $N$-site simplixes. As with the valence bond solids, the new simplex solid (SS) states are extinguished by certain local projection operators, allowing one to construct Hamiltonians with local interactions which render the SS states exact ground states. Using a coherent state representation, I show that the quantum correlations in each SS state are calculable as the finite temperature correlations of an associated classical model with $N$-spin interactions on the same lattice. In three and higher dimensions, the SS states can spontaneously break $\mathrm{SU}(N)$ and exhibit $N$-sublattice long-ranged order as a function of a discrete parameter which fixes the local representation of $\mathrm{SU}(N)$. I analyze this transition using a classical mean field approach. For $N>2$, the ordered state is selected via an ``order by disorder'' mechanism. As in the Affleck-Kennedy-Lieb-Tasaki case, the bulk representations fractionalize at an edge, and the ground state entropy is proportional to the volume of the boundary.

Synthesis, structure and magnetic behaviour of mixed metal leucophosphite

A synthetic analogue of leucophosphite, an iron phosphate, was synthesized hydrothermally at 180 °C and its chemical composition determined to be {K[(FeV)(PO 4) 2(OH)(H 2O)]·H 2O}. The compound crystallizes in the monoclinic P2 1/ c space group, with a=9.7210(19) Å, b=9.6500(19) Å, c=12.198(4) Å and β=128.569(18)°. While the original all-iron compound is reported to be antiferromagnetic, the inclusion of substitutional vanadium(III) ions renders the structure ferrimagnetic. Diffraction studies and magnetic characterization show that iron and vanadium are disordered throughout the crystallographic sites. The magnetic behaviour of this system was interpreted in terms of a classic ferrimagnetic mean field model, consisting of two antiferromagnetically coupled non-crystallographic “sublattices”.

Viscoelastic behavior and electrical properties of flexible nanofiber filled polymer nanocomposites. Influence of processing conditions

Two different types of high aspect ratio flexible nanofibers, cellulose nanofibrils and carbon nanotubes, were dispersed in an amorphous thermoplastic polymer matrix. The mechanical and (in the case of carbon nanotubes filled composites) electrical properties of these composites were investigated. Dynamic mechanical analysis highlighted the influence of entanglements between fibers and of fiber/fiber contact properties on the composite mechanical reinforcement in the rubbery state. In the case of cellulose filled nanocomposites a large mechanical reinforcement effect was observed. This effect was explained by the formation of a rigid nanofibril network linked by strong hydrogen bonds. The formation of this network was assumed to be governed by a percolation mechanism. Conversely, when such bonds between cellulose fibrils were prevented by the process, a lower mechanical reinforcement is observed and can be modeled by a classical mean field approach. On the other hand, both types of composites filled with carbon nanotubes (where no strong interactions are possible) highlighted the fact that entanglements are responsible for a strong increase in thermo-mechanical stability but do not influence the mechanical reinforcement. However, carbon nanotubes are good conductive objects and for such nanocomposites, electrical percolation properties were found. The influence of the process on these electrical properties was highlighted and discussed in term of modification of tube–tube contact electrical properties.

Critical Properties of Weak Ferroelectric LGO Crystals Doped with Cr3+ Ions

Angular diagrams of Cr 3+ EPR spectra in Li 2 Ge 7 O 15 (LGO) crystals confirm that magnetic ions substitute for Ge 4+ and together with interstitial lithium ions form Cr 3+ −Li + paired centers. On approaching ferroelectric phase transition point EPR lines of Cr 3+ broaden and split below T C = 283.4 K. Temperature dependence of the local order parameter η(T) determined from EPR lines splitting, shows power like behavior with critical exponent β = 0.31 of three-dimensional Izing model. Results of previous investigations evidence that on going away from T C the crossover from fluctuation regime to classical mean field one is characteristic for macroscopic and local LGO properties. For dependence η(T) in LGO:Cr such crossover is not detected. Absence of the crossover to classical mean field behavior is attributed to the defect nature of Cr 3+ −Li + centers which weaken the order parameter correlations. Influence of Cr 3+ −Li + centers on LGO properties is discussed on the basis of microscopic model of structural phase transitions.

Relationships Among Liquid–Vapor Interfacial Region Properties: Predictions of a Thermodynamic Model

In theoretical models and molecular dynamics simulations of the interfacial region between a liquid and vapor phase, three properties are usually of primary interest: the interfacial tension, the interfacial region thickness, and the density gradient in the interfacial region. While these properties can be determined from molecular dynamics simulations by collecting appropriate statistics, such results do not explicitly provide an indication of the interrelationship among these characteristics. This paper presents theoretical predictions of the relationships among interfacial tension, the interfacial region thickness, and the density gradient in the interfacial region that are derived from a theoretical model of the thermodynamic properties of the interfacial region. Explicit relations among interfacial region properties are obtained from a modified version of the classical mean field model that incorporates Redlich-Kwong fluid properties. Comparisons are presented that indicate that the theoretical relations among the interfacial region properties are consistent with trends indicated by experimental data. Use of the theoretical model relations to determine the interfacial tension using the mean density profile obtained from a molecular dynamics simulation is also explored. This method is shown to predict values comparable to traditional methods for determining interfacial tension in molecular dynamics simulations while requiring significantly less computational effort.

Numerical experiments in screening theory

In an attempt to clarify the behavior of the screening phenomenon in N-body systems at thermodynamics equi- librium, under the particular conditions at the core of the Sun, we define the screening energy and summarize the two main types of methods to calculate it: (a) by using classical mean field in Statistical Mechanics and (b) ab initio, by using molecular dynamics and first principles only. We invent a pair interaction with a finite parametrized range and strength, and carry out a numerical experiment which demonstrates the difference between the two aforementioned methods and underlying assumptions as a function of the range of the interaction (or equivalently the number of particles in the interaction zone). We study the effect that fluctuations in the number of particles in the screening/neutralizing cloud have on the screening, and the deviations from mean field results which ensue. We also demonstrate that the classical mean field theory is inaccurate under the conditions prevailing in stellar cores and essentially over estimates the screening. We show how the molecular dynamics results tend to the mean field Statistical Mechanics limits when the number of particles in the neutralizing domain increases.

Understanding the Screening of Nuclear Reaction in Stellar Cores

The screening of nuclear reactions in stars takes place under the special conditions that the total number of particles in the screening cloud is of the order of unity. Hence, standard mean field theory is not expected tobe valid. We demonstrate by the use of Molecular dynamics that fluctuations dominate the screening and lead to a smaller screening compared with the predictions of the classical mean field theory.

Ripening processes in supported and pinned nanoclusters—experiment, simulation andtheory

We have investigated experimentally and theoretically the role that ripening processes playin the evolution of supported transition metal nanoclusters on solid surfaces. Wediscuss some avenues of analysis that can be more informative of the dominantmechanisms than mere temporal measures of island size. Our model system isformed by the growth of Pd nanoclusters at room temperature on the cross-linked(1 × 2) reconstructed TiO2(110) surface. The reconstructed surface contains a rectangular array of ‘defects’ that nucleateand pin the clusters. We have followed the surface evolution by variable temperaturescanning tunnelling microscopy as the temperature was raised stepwise to 973 K. We findthat the ripening is dominated by an Ostwald-type mechanism with particles remainingimmobile during growth (decay). Monte Carlo simulations of the ripening of arrays ofthree-dimensional islands have been undertaken for comparison with the experiments.These reproduce the spatial properties of the experimental arrays and the scalingnature of the height distribution. In addition the classic mean field theory for thesize distribution is modified to include island size-separation correlations andis found to recover the size distribution found in both the simulation and theexperiment. We conclude that detailed atomistic understanding of ripening is notalways necessary to understand important features of nanostructure evolution.

Reabsorption effect and spatial ordering in island ripening.

A model for detachment-limited island ripening is analyzed in terms of a reabsorption (or pedophagous) effect where islands releasing monomers can reabsorb their own offspring. When the effect is artificially turned off, ripened island arrays are spatially disordered and the island growth rates and size distribution are easily understood in terms of a classic mean field theory. With the pedophagous effect included, the arrays develop a spatial ordering consistent with experimental studies. The ordering is quantified through a correlation between island size and the number of nearby neighbors, yielding modified mean field island growth rates and a much-improved expression for the island size distribution. It is suggested that the spatial ordering and our explanation are widely applicable to ripening systems.

Thermodynamic properties and structure of the liquid–vapor interface: A neoclassical Redlich–Kwong model

It is well known that the classical mean field theory of van der Waals for liquid–vapor interfacial region properties deviates from real fluid behavior in several important ways. In particular, the variations of the surface tension and interfacial region thickness with temperature near the critical point are not consistent with those for real fluids. This paper presents a modified version of the classical mean field model that incorporates Redlich–Kwong fluid properties. It is shown here that this neoclassical Redlich–Kwong fluid model predicts property variations with temperature that agree better with measured data for real fluids. Predictions of the critical exponents associated with the temperature variation of surface tension and interfacial region thickness are developed from the Redlich–Kwong model. This new model predicts that surface tension varies about proportional to (1−T/Tc)1.33 and that the interfacial region thickness varies about proportional to (1−T/Tc)−0.67, with T and Tc being the system temperature and critical temperature, respectively. Compared to the critical exponents predicted by classical van der Waals theory, the Redlich–Kwong model predictions are found to be in better agreement with values inferred from experimental measurements. Predictions of the Redlich–Kwong model regarding interfacial region structure and property fluctuations are also examined.

The partition function of an interacting many body system

Based on the path integral representation of the partition function of a many body system with separable two body interaction we propose a systematic extension of the perturbed static path approximation (PSPA) to lower temperatures. Thereby, special attention must be paid to instabilities of the classical mean field solution in functional space that cause divergencies within the conventional PSPA. As a result we develop an approximation applicable from high to very low temperatures. These findings are tested against exact results for the archetypical cases of a particle moving in a one dimensional double well and the exactly solvable Lipkin model. In particular, we obtain a very good approximation to the level density of the Lipkin model even at low thermal excitations. Our results may have potential applications in low temperature nuclear physics and mesoscopic systems, e.g. for gap fluctuations in nanoscale superconducting devices previously studied within a PSPA type of approximation. PACS: 5.30.-d, 24.60.-k, 21.10.Ma, 74.25.Bt

Self–similarity and instability in classical mean–field models for domain coarsening

In the classical theory by Lifshitz, Slyozov and Wagner (LSW) coarsening of a dilute system of particles is modelled by a nonlocal transport equation for the particle size distribution. LSW predict that the asymptotic behavior for large times is self–similar and that a particular self–similar profile is approached. In this talk we discuss rigorous results on the long–time behavior of solutions for several variants of this model. For systems in which particle size is uniformly bounded these results establish a sensitive dependence on the data and thus in general do not confirm the predictions by LSW. More precisely we prove that convergence to the classically predicted similarity solution is impossible if the initial distribution is comparable to any finite power of distance to the end of the support. In addition we give a necessary criterion for convergence to other self–similar solutions, which implies non–self–similar asymptotics for a dense set of data.

Phase equilibria of a near-critical ionic system. Critical exponent of the order parameter

The density ρ of coexisting phases of the ternary system water+1,4-dioxane+potassium chloride was investigated along the liquid–liquid–solid coexistence curve near the critical end-point using a vibrating tube densimeter. By visual determination, this lower critical end-point was located at 311.026±0.010 K with a mole ratio dioxane (D) to water (W) rc=nD/nW=0.418±0.004. Density measurements were carried out in the range 0.01 K<(T−Tc)<31.5 K (i.e., 3×10−5<t=(T−Tc)/Tc<0.1). The obtained coexistence curve displays an apparent sharp crossover at reduced temperatures t≃10−2 from a nonclassical (Ising) to a classical (mean field) exponent. In ancillary experiments, the critical salt concentration was determined. Reduced critical values were calculated and compared with those predicted by the simplest ionic model (RPM, Restricted Primitive Model). The relation between critical behavior and reduced critical parameters in the present and other non-Coulombic systems is discussed.

Quantized mean-field approximation

An extension of the quantum–classical mean-field (MF) approximation is developed by application of quantized Hamilton dynamics (QHD) to the classical subsystem. The resulting quantized MF (QMF) approach supplements the classical position and momentum variables with higher order moments. In the limit of all moments exact quantum dynamics is achieved. Already with second order variables, QMF properly treats zero point energy (ZPE) that increases quantum transition probabilities and restricts energy exchange between quantum and classical subsystems. The efficiency of QMF and the effect of ZPE on the transition probability are illustrated by application to a model of O 2 interacting with a Pt surface.

A magnetic study of magnetoresistive Cu–(SmCo 5)–Fe heterogranular alloys

The magnetic interactions between the nanograins in heterogeneous granular Cu–(SmCo 5)–Fe ribbons were evaluated with a classical mean field model to obtain the exchange coupling as well as the magnetic moment per nanoparticle. The samples were subjected to various thermal treatments and a significant magnetoresistive effect was obtained when the optimal annealing conditions had been achieved.

Volume fraction dependence on coarsening of Y 2BaCuO 5 particles in Ba–Cu–O melt

The effect of the volume fraction of the solid phase (Vf) on coarsening phenomena of Y 2BaCuO 5 (Y211) particles in Ba–Cu–O melt has been experimentally investigated. The rate constant of the coarsening cubic-law ( K) also depends strongly on Vf and is classified into two groups. The samples with Vf lower than 15% reveal higher coarsening rate constants than those with Vf higher than 20%. The coarsening properties of the lower Vf were in good agreement with the classical mean field (MF) model that considered the effect of Vf. It was clarified that K values of the higher Vf were suitable for those calculated numerically by the communicating neighbor (CN) model, in which the solute was assumed to diffuse directly among the particles. Thus, the coarsening phenomena of Y211 particles have been qualitatively explained by the difference in coarsening mechanisms as the MF and CN models.

Collision of SU(2) gauge fields in 3+1 dimensions

The Yang–Mills equations provide a classical mean field description of the non-Abelian soft glue field dynamics in heavy ion collisions at high energies. A C++ code for the numerical solution on a three-dimensional gauge lattice is presented. Non-perturbative time-dependent solutions in position space are calculated in the weak coupling regime of a gauge lattice within the Hamiltonian approach of Kogut and Susskind. Applications are demonstrated by simulations of the Yang–Mills field dynamics in collisions of initially uniformly color polarized Gaussian wave packets. The collisions are evolved in time. The time dependent energy densities, energy flows and energy currents in longitudinal and transverse directions with respect to the collision axis are presented. Simulations of this type have recently revealed an interesting solution which exhibits a time-delayed sudden increase of the transverse energy currents in the “non-Abelian case” at short times after the collision.