Random Systems Research Articles

Free-energy fluctuations in a one-dimensional random Ising system

The statistical characteristics of the free-energy fluctuations in a one-dimensional dilute Ising system are investigated by using the fluctuation spectrum theory developed for the global characterization of fluctuations. The rigorous results of free-energy fluctuations for a simplified model are obtained.

Influence of disorder on the interface sharpness of an Ising ferromagnetic system

We study the thermodynamic behavior of a two-component random ferromagnetic Ising system in the presence of given boundary conditions. The system consists of two species A and B occupying the sites of a Bethe lattice which terminates on a surface layer where the spins are kept fixed. We study the interface of the system when the spins on half of the surface of the lattice are fixed opposite to the spins on the other half. More specifically, we study the influence of disorder on the interface width. We find that disorder clearly increases the interface width at temperatures well belowTc, indicating that the interface roughening of disordered Ising systems in 3D real lattices should occur at temperatures significantly below those of the corresponding ordered ones.

TRANSFER TENSOR VARIATION METHOD APPLIED TO RANDOM ISING SYSTEMS

TRANSFER TENSOR VARIATION METHOD APPLIED TO RANDOM ISING SYSTEMS

One-dimensional random Ising systems with interaction decayr ?(1+?): A convergent cluster expansion

WE consider a one-dimensional random Ising model with Hamiltonian $$H = \sum\limits_{i\ddag j} {\frac{{J_{ij} }}{{\left| {i - j} \right|^{1 + \varepsilon } }}S_i S_j } + h\sum\limits_i {S_i } $$ , where e>0 andJ ij are independent, identically distributed random variables with distributiondF(x) such that $$\int {xdF\left( x \right) = 0} $$ (i)) , $$\int {e^{tx} dF\left( x \right)< \infty \forall t \in \mathbb{R}} $$ (ii)) . We construct a cluster expansion for the free energy and the Gibbs expectations of local observables. This expansion is convergent almost surely at every temperature. In this way we obtain that the free energy and the Gibbs expectations of local observables areC∞ functions of the temperature and of the magnetic fieldh. Moreover we can estimate the decay of truncated correlation functions. In particular for every e′>0 there exists a random variablec(ω)m, finite almost everywhere, such that $$\left| {\left\langle {s_0 s_j } \right\rangle _H - \left\langle {s_0 } \right\rangle _H \left\langle {s_j } \right\rangle _H } \right| \leqq \frac{{c\left( \omega \right)}}{{\left| j \right|^{1 + \varepsilon - \varepsilon '} }}$$ , where 〈 〉 H denotes the Gibbs average with respect to the HamiltonianH.

Statistical mechanics of Eigen's evolution model

The correspondence between Eigen's model of macromolecular evolution and the equilibrium statistical mechanics of an inhomogeneous Ising system is developed. The free energy landscape of random Ising systems with the Hopfield Hamiltonian as a special example is applied to the replication rate coefficient landscape. The coupling constants are scaled with 1/l, since the maxima of any landscape must not increase with the length of the macromolecules. The calculated error threshold relation then agrees with Eigen's expression, which was derived in a different way. It gives an explicit expression for the superiority parameter in terms of the parameters of the landscape. The dynamics of selection and evolution is discussed.

Dynamics of droplet fluctuations in pure and random Ising systems

Long-lived droplet fluctuations can dominate the long-time equilibrium dynamics of long-range-ordered Ising systems, yielding nonexponential decay of temporal spin autocorrelations. For the two-dimensional pure Ising model the long-time decay is a stretched exponential, exp(- \ensuremath{\surd}t/\ensuremath{\tau} ), where t is time and \ensuremath{\tau} a correlation time. For systems with quenched random-exchange disorder the spatially averaged correlation decays as a power of time, ${t}^{\mathrm{\ensuremath{-}}x}$, with the exponent x in general being nonuniversal. For systems with quenched random-field disorder the decay is slower still, as exp[-k(lnt${)}^{(d\mathrm{\ensuremath{-}}2)/(d\mathrm{\ensuremath{-}}1)}$], where k is a nonuniversal number and d is the dimensionality of the system. The low-frequency noise from this slow dynamics may be experimentally detectable, as is the analogous noise in spin-glass ordered phases.

Phase transitions in diluted magnets: Critical behavior, percolation, and random fields

Transition metal halides provide realizations of Ising,XY, and Heisenberg antiferromagnets in one, two, and three dimensions. The interactions, which are of short range, are generally well understood. By dilution with nonmagnetic species such as Zn++ or Mg++ one is able to prepare site-random alloys which correspond to random systems of particular interest in statistical mechanics. By mixing two magnetic ions such as Fe++ and Co++ one can produce magnetic crystals with competing interactions-either in the form of competing anisotropies or competing ferromagnetic and antiferromagnetic interactions. In this paper the results of a series of neutron scattering experiments on these systems carried out at Brookhaven over the past several years are briefly reviewed. First the critical behavior in Rb2Mn0.5Ni0.5F4 and FecZn1−cF2 which correspond to two-dimensional and three-dimensional random Ising systems, respectively, are discussed. Percolation phenomena have been studied in Rb2MncMgl−cF4, Rb2CocMgl−cF4, KMncZl-cF3, and MncZnl−cF2 which correspond to two-and three-dimensional Heisenberg and Ising models, respectively. In these casesc is chosen to be in the neighborhood of the nearest-neighbor percolation concentration. Application of a uniform field to the above systems generates a random staggered magnetic field; this has facilitated a systematic study of the random field problem. As we shall discuss in detail, a variety of novel, unexpected phenomena have been observed.

Convergence of the Cluster Variation Method for Random Ising Models in the Thermodynamic Limit

Schlijper (Phys. Rev. B27 (1983), 6841) studied the variational principle in the cluster variation method of cooperative phenomena and proved that the values of the free energy obtained in successive approximations of that method converge to the exact free energy in the thermodynamic limit. His result is extended to the random Ising model. 2 ) proved that, if we choose an appropriate sequence of approximations, the successive values of the free energy obtained in that method converge to the exact value. In the present paper, we show that the same is true also for the random Ising model. Schlijper's arguments start with the variational principle in the thermodynamic limit. In order to give the corresponding variational principle for the random system, we recall the variational principle developed for finite random Ising systems by the present author,3) in §2. In §3, we give the variational principle in the thermodynamic limit. We then see that Schlijper's arguments apply to the present case with slight modification, in §§3 and 4. In the Appendix, Schlijper's lemma 1 is discussed.

Study of two-component random Ising systems including site correlations

Study of two-component random Ising systems including site correlations

Influence of clustering interactions on the magnetic properties of two-component random Ising systems

The authors study the positional (site) correlation of the magnetic 'atoms' in a two-component random Ising system, induced in the liquid state by attractive or repulsive interactions U among them. This site correlation strongly influences the magnetic properties of the system. The Tcr versus concentration phase diagrams of such systems are studied for various values of the interaction U and the results compared to observe phase diagrams of relevant real systems.

Study of two-component random Ising systems including site correlations

The authors study the thermodynamic behaviour of two-component random Ising systems in the formalism of Eggarter and Eggarter (1977) properly extended to include site correlations. This formalism is exact in the case of a Bethe lattice, while it leads to the Bethe-Peierls approximation for real lattices. The authors consider such systems with antiferromagnetic couplings, different values of the magnetic moments and placed in a magnetic field. They calculate their magnetic properties and discuss them in connection with relevant experimental studies. They also study the phase diagrams of such systems, concentrating on cases relevant to other theoretical as well as experimental studies.

Critical amplitudes for the random Ising system

The critical amplitude ratios C+/C-, A+/A- and f+/f- are calculated to one loop and compared with their experimental values.

Phase diagrams of random Ising models : simple and systematic successive approximations

Abstract 2014 We present a simple and systematic method to calculate phase diagrams of random Ising models. It applies to a wide class of systems in which, however, randomness has to be described by discrete random varia-bles. We calculated, at different orders of approximation, various quantities like critical temperatures and perco- lation thresholds in good agreement with exact known results. J. Physique 44 (1983) 1143-1147 OCTOBRE 1983, : Classification Physics Abstracts 64.60 Most papers dealing with random Ising systems make use either of real space renormalization group methods [1] or the replica trick combined with themean-field approximation [2]. This paper presents a much less sophisticated method due to one of us [3] which is, conceptually, as simple as an ordinary mean- field approximation and has the advantage of leading to systematic successive approximations which, even at the lowest order, give, from a numerical point of view, better results than the mean-field

Griffiths singularity in ferromagnetic alloys

Griffiths's result for percolating Ising ferromagnets is extended to the case of binary ferromagnetic alloys. The results indicate that the Griffiths singularity is a general and characteristic phenomenon in random Ising systems. This lends strong support to the contention of McCoy and Wu that a single length-scaling picture is not adequate to describe phase transitions in quenched random magnets.

Concentration Expansion for Random Ising Systems. II

Two-dimensional Ising systems with bond-impurities are treated exactly by using the method of Gibberd and Hurst. First we derive recurrence equations by which a Green's function with n scattering centers can be obtained from that with n -1 scattering centers. Using these recurrence equations, we are able to expand the free energy systematically in power series of c'oncentration c of impure bonds. This concentration expansion is carried out up to the order of c' for quenched and annealed systems, and we discuss critical properties of the quenched system comparing with those of the annealed system. The difference between the two systems is seen to arise first in the term of the order c', but the singularities in these terms are almost the same for both systems. This result leads to an conjecture that critical properties of a quenched system are the same as those of an annealed system in the critical region. The shift of the critical temperature in the quenched system is also calculated up to the order of c' in the bond-dilution case (the strength of the impurity bonds is zero).

Concentration Expansion for Random Ising Systems

Two-dimensional Ising systems with bond-impurities are treated exactly by using the method of Gibberd and Hurst. First we derive recurrence equations by which a Green's function with n scattering centers can be obtained from that with n-1 scattering centers. Using these recurrence equations, we are able to expand the free energy systematically in power series of concentration c of impure bonds. This concentration expansion is carried out up to the order of c2 for quenched and annealed systems, and we discuss critical properties of the quenched system comparing with those of the annealed system. The difference between the two systems is seen to arise first in the term of the order c2, but the singularities in these terms are almost the same for both systems. This result leads to an conjecture that critical properties of a quenched system are the same as those of an annealed system in the critical region. The shift of the critical temperature in the quenched system is also calculated up to the order of c2 in the bond-dilution case (the strength of the impurity bonds is zero).

Frustration effect on the d-dimensional Ising spin glass. I. Spin glass and dilution problems

The d-dimensional random bond Ising system is treated by the square approximation generalised for a random system. The expression of the uniform, staggered, and spin glass susceptibilities is obtained for the general distribution of the exchange interaction. The present approximation includes the closed-loop and frustration effects directly, the pure limit of which agrees with Kikuchi's second approximation. A simple version of the above approximation is applied to the spin glass and dilution problem and the authors obtain the phase diagram of the ferromagnetic, spin glass and paramagnetic phases including the frustration effect.

Cluster extension of the effective-interaction approximation. I. Quenched-bond disorder in the Ising model

A systematic cluster extension of the effective-interaction approximation for quenched-bond disorder in random Ising systems is proposed. For bond dilution, exact results for the critical concentrations are obtained in the single-bond approximation for the self-dual square lattice and in the three-bonds approximation for the dual triangular and honeycomb lattices. Qualitative changes in the critical line are observed in the frustration model when going from the single-bond to the three-bonds approximations.

On the random binary Ising system with a transverse field

The transition temperature and paramagnetic susceptibility in the random binary Ising model with a transverse field are obtained. The system is composed of ferro- and antiferromagnetic spins. It is found that ferro- and antiferromagnetic states coexist in some cases.

Phase transitions in the frustration network

The distribution of the frustrated plaquettes is calculated as a function of disorder for two and three dimensional random Ising- andXY-models. It is shown that in all cases the frustrated plaquettes are highly correlated and that at a critical value of disorder a phase transition occurs in the frustration network which can be viewed as a dissociation of plaquettes. The critical disorder is calculated using duality transformations. We show that in the two and three dimensional random Ising systems two types of nonmagnetic phases exist which differ qualitatively by their distribution of frustrated plaquettes and consequently by their domain wall structure.