Showing papers on "Ising model published in 1985"

Introduction to percolation theory

Abstract: Preface to the Second Edition Preface to the First Edition Introduction: Forest Fires, Fractal Oil Fields, and Diffusion What is percolation? Forest fires Oil fields and fractals Diffusion in disordered media Coming attractions Further reading Cluster Numbers The truth about percolation Exact solution in one dimension Small clusters and animals in d dimensions Exact solution for the Bethe lattice Towards a scaling solution for cluster numbers Scaling assumptions for cluster numbers Numerical tests Cluster numbers away from Pc Further reading Cluster Structure Is the cluster perimeter a real perimeter? Cluster radius and fractal dimension Another view on scaling The infinite cluster at the threshold Further reading Finite-size Scaling and the Renormalization Group Finite-size scaling Small cell renormalization Scaling revisited Large cell and Monte Carlo renormalization Connection to geometry Further reading Conductivity and Related Properties Conductivity of random resistor networks Internal structure of the infinite cluster Multitude of fractal dimensions on the incipient infinite cluster Multifractals Fractal models Renormalization group for internal cluster structure Continuum percolation, Swiss-cheese models and broad distributions Elastic networks Further reading Walks, Dynamics and Quantum Effects Ants in the labyrinth Probability distributions Fractons and superlocalization Hulls and external accessible perimeters Diffusion fronts Invasion percolation Further reading Application to Thermal Phase Transitions Statistical physics and the Ising model Dilute magnets at low temperatures History of droplet descriptions for fluids Droplet definition for the Ising model in zero field The trouble with Kertesz Applications Dilute magnets at finite temperatures Spin glasses Further reading Summary Numerical Techniques

Interacting Particle Systems

Abstract: The Construction, and Other General Results.- Some Basic Tools.- Spin Systems.- Stochastic Ising Models.- The Voter Model.- The Contact Process.- Nearest-Particle Systems.- The Exclusion Process.- Linear Systems with Values in [0, ?)s.

Spin-glass models of neural networks.

Abstract: Two dynamical models, proposed by Hopfield and Little to account for the collective behavior of neural networks, are analyzed. The long-time behavior of these models is governed by the statistical mechanics of infinite-range Ising spin-glass Hamiltonians. Certain configurations of the spin system, chosen at random, which serve as memories, are stored in the quenched random couplings. The present analysis is restricted to the case of a finite number p of memorized spin configurations, in the thermodynamic limit. We show that the long-time behavior of the two models is identical, for all temperatures below a transition temperature ${T}_{c}$. The structure of the stable and metastable states is displayed. Below ${T}_{c}$, these systems have 2p ground states of the Mattis type: Each one of them is fully correlated with one of the stored patterns. Below T\ensuremath{\sim}0.46${T}_{c}$, additional dynamically stable states appear. These metastable states correspond to specific mixings of the embedded patterns. The thermodynamic and dynamic properties of the system in the cases of more general distributions of random memories are discussed.

Pinning and roughening of domain walls in Ising systems due to random impurities.

Abstract: Randomly placed impurities that alter the local exchange couplings, but do not generate random fields or destroy the long-range order, roughen domain walls in Ising systems for dimensionality $\frac{5}{3}ldl5$. They also pin (localize) the walls in energetically favorable positions. This drastically slows down the kinetics of ordering. The pinned domain wall is a new critical phenomenon governed by a zero-temperature fixed point. For $d=2$, the critical exponents for domain-wall pinning energies and roughness as a function of length scale are estimated from numerically generated ground states.

Transfer-matrix method and Monte Carlo simulation in quantum spin systems

Abstract: Transfer-matrix methods for quantum spin systems are formulated and their limiting properties are studied rigorously. The present formulation is applied explicitly to an exactly soluble transverse Ising model. A computer implementation of the two-dimensional triangular antiferromagnetic quantum Heisenberg model is also proposed to study Anderson's picture of the dynamic coherence of the phase of singlet pairs.

Phase diagrams for dilute spin glasses

Abstract: A generalised, dilute, infinite-ranged Ising spin-glass model is introduced and studied as a function of the concentration p and temperature T. The phase diagram is investigated and paramagnetic (P), ferromagnetic (F), spin glass (SG) and mixed (M) phases, meeting at a multicritical point (p*,T*), are identified. The P/F and P/SG phase boundaries are derived, and the F/M and M/SG boundaries are calculated close to (p*,T*). The condition for having a re-entrant spin-glass transition is derived. In non-zero magnetic field a p-dependent A-T instability line is obtained. The authors apply their results to the insulator EuxSr1-xS, it is predicted to exhibit re-entrant behaviour.

Critical behavior of three-dimensional Ising spin-glass model

Abstract: First results of massive Monte Carlo simulations of the d=3 Ising spin-glass with ±J bond distribution, performed on a fast special purpose computer, are presented. A qualitative change in the behavior of the system and best fits for the spin-glass correlation length and relaxation time favor equilibrium phase transition at Tc/J≊1.2. .AE

Facilitated kinetic Ising models and the glass transition

Abstract: The class of ‘‘facilitated’’ kinetic Ising models introduced in a recent letter is investigated in greater detail An interpretation of the models in terms of relaxation processes in dense fluids is described Various types of kinetic behavior are predicted for the different spin models: (A) Arrhenius temperature dependence of the average structural relaxation time, (B) non‐Arrhenius temperature dependence with a divergence of the relaxation time at a nonzero temperature, and (C) non‐Arrhenius temperature dependence with a divergent relaxation time only at zero temperature All of the models show nonexponential decay of equilibrium time correlation functions, consistent with the Kohlrausch–Williams–Watts empirical form The nature of the glass transitions exhibited by the various models is discussed

Model calculations for wetting transitions in polymer mixtures

Abstract: Partially compatible binary mixtures of linear flexible polymers are considered in the presence of a wall which preferentially adsorbs one component. Using a Flory-Huggins type mean field approach, it is shown that in typical cases at two-phase coexistence the wall is always « wet », i.e. coated with a macroscopically thick layer of the preferred phase, and the transition to the non wet state occurs at volume fractions of the order of 1/~N (where N is the chain length) at the coexistence curve. Both first and second order wetting transitions are found, and the variation of the surface layer thickness, surface excess energy and related quantities through the transition is studied. We discuss both the validity of the long wavelength approximation involved in our treatment, and pos- sible fluctuation effects for « critical wetting », comparing our results to Monte Carlo simulations of wetting in Ising models. The relation of our results to previous work and possible experimental consequences are also briefly mentioned.

Percolation in interacting colloids.

Abstract: The percolation behavior of spherical particles with attractive interactions is studied with use of Monte Carlo simulations. These systems differ from lattice Ising systems, which have been previously analyzed, in the necessity to define a shell parameter \ensuremath{\delta} to specify a connected cluster. For small values of \ensuremath{\delta}, correlations due to the attractive interactions drastically lower the percolation threshold in the vicinity of the gas-liquid critical point. For larger values of \ensuremath{\delta}, these shifts are smaller, but the effects of long-range correlations show up as enhanced finite-size effects. The simulation results are discussed in the light of recent experiments which measure the temperature and concentration dependence of the conductivity in interacting microemulsions.

Finite-Size Tests of Hyperscaling

Abstract: The possible form of hyperscaling violations in finite-size scaling theory is discussed. The implications for recent tests in Monte Carlo simulations of the d = 3 Ising model are examined, and new results for the d = 5 Ising model are presented.

Conformal Invariance, Unitarity and Two-Dimensional Critical Exponents

Abstract: We show that conformai invariance and unitarity severely limit the possible values of critical exponents in two dimensional systems by finding the discrete series of unitarisable representations of the Virasoro algebra. The realization of conformai symmetry in a given system is parametrized by a real number c, the coefficient of the trace anomaly. For c<1 the only values allowed by unitarity are c=1–6/m(m+1), m=2,3,4 ⋯ . For each of these values of c unitarity determines a finite set of rational numbers that must contain all possible critical exponents. These finite sets account for the known critical exponents of the following two dimensional models: Ising(m=3), tricritical Ising(m=4), 3-state Potts(m=5), and tricritical 3-state Potts(m=6).

Scaling theory of the random-field Ising model

Abstract: The scaling laws derived by Grinstein (1976) for the random-field Ising model (RFIM) are rederived on the assumption that the transition is second order and that the critical behaviour is controlled by a zero-temperature fixed point The scaling laws involve three independent exponents nu , eta and gamma , the last appearing in a modified hyperscaling relation, 2- alpha =(d-y) nu It is argued that such hyperscaling modifications are a general feature of phase transitions controlled by zero-temperature fixed points Explicit evaluation of the RFIM exponents in d=2+ epsilon dimensions, yields, to order epsilon , 1/ nu = epsilon , eta =1- epsilon /2 and y=1+ epsilon /2 The exponent nu is different from that of the pure model in (d-y) dimensions implying that no exact 'dimensional reduction' is possible near two dimensions

Quantum tunneling with dissipation and the Ising model over ℝ

Abstract: We consider a quantum particle in a double-well potential, for simplicity in the two-level approximation, coupled to a phonon field. We show that static and dynamical ground state correlations of the particle and of the field are expressible through expectations in an Ising model over ℝ (rather than ℤ). Its free measure is a spin flip process with flip rate ɛ, the difference in energy between the ground state and the first excited state. The Ising model has a ferromagnetic pair interaction whose form depends on the couplings to the phonon field and on the dispersion relation of the phonon field. In physical applications the interaction is long ranged and decays ast −2 for large distances. In this case we prove that for sufficiently strong coupling the particle becomes localized in one of the wells. The effective tunnel rate is zero. The transition to localization is associated with the generation of an infinite number of low momentum phonons. We apply the Ising technology to our problem and discuss the phase diagram in some detail.

Uniaxial Anisotropy Effects in the Ising Model: an Exactly Soluble Model

Abstract: The effect of the uniaxial anisotropy in an Ising model consisting of mixed spins of magnitudes s and 1/2, on the honeycomb lattice, is studied. The exact solution is determined by mapping the model onto an effective spin 1/2 Ising model on the triangular lattice. The transition temperature is obtained as a function of the parameters and it is shown that for s integer there is no long range order for the parameter values in a determined region which is independent of the value of s. The critical exponents are obtained and it is shown that they are the same as those of the two-dimensional spin 1/2 Ising model.

Coexistence of spin-glass and antiferromagnetic orders in the Ising system Fe0.55Mg0.45Cl2.

Abstract: The low-temperature phase of the dilute Ising antiferromagnet ${\mathrm{Fe}}_{0.55}$${\mathrm{Mg}}_{0.45}$${\mathrm{Cl}}_{2}$ in zero field is studied by specific-heat, ac-susceptibility, and neutron-scattering experiments. We find that in this phase, spin-glass behavior and antiferromagnetic long-range order coexist. Such a mixed-order phase is predicted by the mean-field theory of reentrant spin-glasses, but has never been observed before.

Thermodynamical properties of the transverse Ising model

Abstract: A new effective field theory is proposed and used to derive the thermodynamical properties of the transverse Ising model. The formalism is based on an exact formal spin identity for the two-state transverse Ising model and utilizes an exponential operator technique. The method, which can explicitly and systematically include correlation effects, is illustrated in several lattice structures by employing its simplest approximation version (in which spin-spin correlations are neglected). The lines of critical points in the Ω- T plane as well as the thermal behaviour of both transverse and longitudinal magnetizations are analysed for square and simple cubic lattices. It is shown that the present formalism, in spite of its simplicity, yields results which represent a remarkable improvement on the standard mean field treatment (MFA).

The Ground State of the Three-Dimensional Random-Field Ising Model

Abstract: We prove that the three-dimensional Ising model in a random magnetic field exhibits long-range order at zero temperature and small disorder. Hence the lower critical dimension for this model is two (or less) and not three as has been suggested by some.

An Upper Bound on the Critical Percolation Probability for the Three- Dimensional Cubic Lattice

Abstract: We prove that the critical probability for site percolation on the three-dimensional cubic lattice satisfies the inequality $p^{(3)}_c < 1/2$. An application to the three-dimensional Ising model is given.

Strings in two-dimensional classical XY models.

Abstract: We study a class of simple, translationally invariant, two-dimensional, nearest-neighbor, isotropic XY models which possess, in addition to the familiar integer vortices, half-integer vortex and ``string'' excitations. The half-integer vortices interact through both the logarithmic Coulomb potential and a linear potential mediated by the strings. The phase diagram of these models consists of three phases separated by lines of conventional Kosterlitz-Thouless transitions and lines of Ising transitions driven by the vanishing of the tension in the strings.

Interface moving through a random background.

Abstract: We study the motion of a phase interface which is driven through a random background medium, with application to immiscible-fluid displacement in porous media and to random-field Ising systems. The interface motion is described by a local stochastic differential equation, with terms corresponding to an external driving force, interface elasticity, and a random background force. The same equation has been examined by Bruinsma and Aeppli, with whose conclusions we disagree in part. In mean-field theory, we find that the interface can either translate with constant velocity and average width, or be pinned by the random background. The pinning is associated with invasion percolation in the fluid-displacement application and with metastable domains in the Ising case. Perturbation theory in the random term is consistent with the mean-field behavior above three dimensions but diverges in time at lower dimensions, suggesting a transition. The perturbation series appears to be unrenormalizable. By numerical integration of the differential equation, we find that in dimensions less than or equal to 3, the interface pins at sufficiently large randomness but translates essentially as a plane otherwise, while in four dimensions the interface always translates and is never pinned.

Statistical mechanical methods in particle structure analysis of lattice field theories. II. Scalar and surface models

Abstract: We illustrate on simple examples a new method to analyze the particle structure of lattice field theories. We prove that the two-point function in Ising and rotator models has an Ornstein-Zernike correction at high temperature. We extend this to Ising models at low temperatures if the lattice dimensiond≧3. We prove that the energy-energy correlation function at high temperatures (for Ising orN=2 rotators) decays according to mean field theory (i.e. with the square of the Ornstein-Zernike correction) ifd≧4. We also study some surface models mimicking the strong-coupling expansion of the glueball correlation function. In the latter model, besides Ornstein-Zernike decay, we establish the presence of two nearly degenerate bound states.

Critical Ising spin dynamics on percolation clusters.

Abstract: Des spins d'Ising en interaction ferromagnetique sont places sur un reseau fractal (tel qu'un cluster de percolation) avec T c =0, et sont dotes d'une dynamique simple de retournement de spin. A basse temperature, la dynamique collective est determinee par l'activation thermique au-dessus des barrieres d'energie. La barriere associee a la formation de domaines de taille L est proportionnelle a ZlnL ou Z est un nouveau parametre geometrique caracterisant le fractal