Showing papers on "Ising model published in 1993"
TL;DR: A randomised algorithm which evaluates the partition function of an arbitrary ferromagnetic Ising system to any specified degree of accuracy is presented.
Abstract: The paper presents a randomised algorithm which evaluates the partition function of an arbitrary ferromagnetic Ising system to any specified degree of accuracy. The running time of the algorithm in...
660 citations
TL;DR: The zero-temperature random-field Ising model is used to study hysteretic behavior at first-order phase transitions using mean-field theory and results of numerical simulations in three dimensions are presented.
Abstract: We use the zero-temperature random-field Ising model to study hysteretic behavior at first-order phase transitions. Sweeping the external field through zero, the model exhibits hysteresis, the return-point memory effect, and avalanche fluctuations. There is a critical value of disorder at which a jump in the magnetization (corresponding to an infinite avalanche) first occurs. We study the universal behavior at this critical point using mean-field theory, and also present results of numerical simulations in three dimensions.
518 citations
TL;DR: In this article, the conceptual foundations of the renormalization-group (RG) formalism are considered and rigorous theorems on the regularity properties and possible pathologies of the RG map are presented.
Abstract: We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Our main results apply to local (in position space) RG maps acting on systems of bounded spins (compact single-spin space). Regarding regularity, we show that the RG map, defined on a suitable space of interactions (=formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the pathological side, we make rigorous some arguments of Griffiths, Pearce, and Israel, and prove in several cases that the renormalized measure is not a Gibbs measure for any reasonable interaction. This means that the RG map is ill-defined, and that the conventional RG description of first-order phase transitions is not universally valid. For decimation or Kadanoff transformations applied to the Ising model in dimension d⩾3, these pathologies occur in a full neighborhood {β>β0, ¦h¦
488 citations
Book•
01 Jan 1993
TL;DR: In this paper, the authors present a method for the computation of the Lyapunov exponent of PRM in the context of one-dimensional ising models and localization in two and three dimensions.
Abstract: I Background.- 1. Why Study Random Matrices?.- 1.1 Statistics of the Eigenvalues of Random Matrices.- 1.1.1 Nuclear Physics.- 1.1.2 Stability of Large Ecosystems.- 1.1.3 Disordered Harmonic Solids.- 1.2 Products of Random Matrices in Chaotic and Disordered Systems.- 1.2.1 Chaotic Systems.- 1.2.2 Disordered Systems.- 1.3 Some Remarks on the Calculation of the Lyapunov Exponent of PRM.- 2. Lyapunov Exponents for PRM.- 2.1 Asymptotic Limits: the Furstenberg and Oseledec Theorems.- 2.2 Generalized Lyapunov Exponents.- 2.3 Numerical Methods for the Computation of Lyapunov Exponents.- 2.4 Analytic Results.- 2.4.1 Weak Disorder Expansion.- 2.4.2 Replica Trick.- 2.4.3 Microcanonical Method.- II Applications.- 3. Chaotic Dynamical Systems.- 3.1 Random Matrices and Deterministic Chaos.- 3.1.1 The Independent RM Approximation.- 3.1.2 Independent RM Approximation: Perturbative Approach.- 3.1.3 Beyond the Independent RM Approximation.- 3.2 CLE for High Dimensional Dynamical Systems.- 4. Disordered Systems.- 4.1 One-Dimensional Ising Model and Transfer Matrices.- 4.2 Random One-Dimensional Ising Models.- 4.2.1 Ising Chain with Random Field.- 4.2.2 Ising Chain with Random Coupling.- 4.3 Generalized Lyapunov Exponents and Free Energy Fluctuations.- 4.4 Correlation Functions and Random Matrices.- 4.5 Two-and Three-Dimensional Systems.- 5. Localization.- 5.1 Localization in One-Dimensional Systems.- 5.1.1 Exponential Growth and Localization: The Borland Conjecture.- 5.1.2 Density of States in One-Dimensional Systems.- 5.1.3 Conductivity and Lyapunov Exponents: The Landauer Formula.- 5.2 PRMs and One-Dimensional Localization: Some Applications.- 5.2.1 Weak Disorder Expansion.- 5.2.2 Replica Trick and Microcanonical Approximation.- 5.2.3 Generalized Localization Lengths.- 5.2.4 Random Potentials with Extended States.- 5.3 PRMs and Localization in Two and Three Dimensions.- 5.4 Maximum Entropy Approach to the Conductance Fluctuations.- III Miscellany.- 6. Other Applications.- 6.1 Propagation of Light in Random Media.- 6.1.1 Media with Random Optical Index.- 6.1.2 Randomly Deformed Optical Waveguide.- 6.2 Random Magnetic Dynamos.- 6.3 Image Compression.- 6.3.1 Iterated Function System.- 6.3.2 Determination of the IFS Code for Image Compression.- 7. Appendices.- 7.1 Statistics of the Eigenvalues of Real Random Asymmetric Matrices.- 7.2 Program for the Computation of the Lyapunov Spectrum.- 7.3 Poincare Section.- 7.4 Markov Chain and Shannon Entropy.- 7.5 Kolmogorov-Sinai and Topological Entropies.- 7.6 Generalized Fractal Dimensions and Multifractals.- 7.7 Localization in Correlated Random Potentials.- References.
346 citations
TL;DR: In this article, a review is given of the differential operator technique in the Ising spin systems and theoretical frameworks of the various models are discussed on the basis of the ISing spin identities.
Abstract: A review is given of the differential operator technique in the Ising spin systems. The theoretical frameworks of the various models are discussed on the basis of the Ising spin identities. These can be applied to examine the magnetic properties in a variety of magnetic materials.
291 citations
TL;DR: A modification of the Metropolis Monte Carlo scheme in sequence space with an evolutionary temperature which sets the energy scale is proposed, implying that the design algorithm does not encounter multiple-minima problems and is very fast.
Abstract: We propose a simple algorithm to design a sequence which fits a given protein structure with a given energy. The algorithm is a modification of the Metropolis Monte Carlo scheme in sequence space with an evolutionary temperature which sets the energy scale. There is a one to one correspondence between this optimization scheme and the Ising model of ferromagnetism. This analogy implies that the design algorithm does not encounter multiple-minima problems and is very fast. The algorithm is tested by «predicting» the primary structures of four proteins. In each case the calculated primary structures had statistically significant homology with the natural structures
237 citations
Book•
01 Jan 1993
TL;DR: Path Integrals and Quantum Mechanics Harmonic Oscillator Generating Functional Path Integrals for Fermions Supersymmetry Semi-Classical Methods Path Integral for the Double Well Path integral for Relativistic Theories Effective Action Invariances and their Consequences Gauge Theories Anomalies Systems at Finite Temperature Ising Model as mentioned in this paper.
Abstract: Path Integrals and Quantum Mechanics Harmonic Oscillator Generating Functional Path Integrals for Fermions Supersymmetry Semi-Classical Methods Path Integral for the Double Well Path Integral for Relativistic Theories Effective Action Invariances and Their Consequences Gauge Theories Anomalies Systems at Finite Temperature Ising Model.
161 citations
TL;DR: In this paper, the authors derived the phenomenological dynamics of interfaces from stochastic "microscopic" models and derived Green-Kubo-like expressions for the mobility.
Abstract: We derive the phenomenological dynamics of interfaces from stochastic “microscopic” models. The main emphasis is on models with a nonconserved order parameter. A slowly varying interface has then a local normal velocity proportional to the local mean curvature. We study bulk models and effective interface models and obtain Green-Kubo-like expressions for the mobility. Also discussed are interface motion in the case of a conserved order parameter, pure surface diffusion, and interface fluctuations. For the two-dimensional Ising model at zero temperature, motion by mean curvature is established rigorously.
149 citations
TL;DR: In this paper, the universal critical point ratios of the second and fourth moments of the magnetization for ferromagnetic Ising models on the square and on the triangular lattices were calculated.
Abstract: The authors calculate the universal critical-point ratios of the square of the second and the fourth moment of the magnetization for ferromagnetic Ising models on the square and on the triangular lattices. Periodic boundary conditions are used in accordance with the four-fold and six-fold rotational symmetries of the respective lattices. These results, which are obtained by means of an analysis of finite-size data computed with a transfer-matrix technique, have an accuracy of the order of one millionth. This analysis is also applied to rectangular systems with arbitrary aspect ratios.
143 citations
TL;DR: Dynamic scaling of magnetic hysteresis in a few monolayer thick Fe/Au(001) films was studied by using surface magneto-optic Kerr effect and confirmed a recently proposed scaling law.
Abstract: Dynamic scaling of magnetic hysteresis in a few monolayer thick Fe/Au(001) films was studied by using surface magneto-optic Kerr effect. For low values of the frequency (Ω) and amplitude (H 0 ) of an applied magnetic field, the hysteresis loss scales as A∞H 0 a Ω β with α=0.59±0.07 and β=0.31±0.05. Our results confirm a recently proposed scaling law. The exponents are consistent with recent numerical simulations of the hysteresis of 2D Ising spins and suggest that the hysteresis dynamics in ultrathin ferromagnetic films belong to a dynamic Ising universality class
135 citations
TL;DR: In this article, the Harris criterion is applied to aperiodic structures with unbounded geometrical fluctuations, whenever their wandering exponent exceeds a model-dependent threshold value, which explains why crystals and quasi-crystals exhibit the same critical behaviour.
Abstract: We investigate statistical-mechanical models on aperiodic structures (e.g. quasi-periodic, random, self-similar). We derive a condition for the relevance of any kind of aperiodicity near a continuous phase transition, in the spirit of the Harris criterion. This explains why crystals and quasi-crystals exhibit the same critical behaviour. Novel universality classes of critical phenomena are predicted on structures with unbounded geometrical fluctuations, whenever their wandering exponent exceeds a model-dependent threshold value.
TL;DR: The Ising model has been used very successfully in the computation of alloy phase diagrams and it is shown that this success is not coincidental by deriving the model from the full partition function of the alloy.
TL;DR: It is concluded that, contrary to theoretical expectations, quantum transitions can be qualitatively different from thermally driven transitions in real spin glasses.
Abstract: LiHo_(0.167)Y_(0.833)F_4 is a dilute dipolar-coupled Ising magnet with a spin glass transition which can be crossed with temperature T (T_g=0.13 K) or with an effective transverse field Γ(Γ_g=1 K at T=0). The nonlinear susceptibility contains a diverging component which dominates at T=98 mK, but disappears by 25 mK. At the same time, the onset of spin glass behavior in the dissipative linear susceptibility becomes sharper. We conclude that, contrary to theoretical expectations, quantum transitions can be qualitatively different from thermally driven transitions in real spin glasses.
TL;DR: In this paper, critical exponents of chiral Ising, XY and Heisenberg universality classes up to second order in 4 − ϵ expansion were calculated and compared with recent extensive 1/N calculation and lattice simulations.
TL;DR: In this article, the properties of a family of nonequilibrium spin models with up-down symmetry on a square lattice are determined by a mean-field pair approximation and by Monte Carlo simulation.
Abstract: The properties of a family of nonequilibrium spin models with up-down symmetry on a square lattice are determined by a mean-field pair approximation and by Monte Carlo simulation. The phase diagram in the parameter space displays a critical line that terminates at a first-order critical point. It is found that the critical exponents are the same as those of the equilibrium Ising model.
TL;DR: In this article, the mean field solution of the general Blume-Capel model with integer and semi-integer spins was studied and a multiphase point in which the different phases spread out when the temperature is increased was found.
TL;DR: The finite-size scaling results for the critical exponents yield the two-dimensional Ising-model values, which are in good agreement with those suggested by the universality hypothesis.
Abstract: The single-spin-flip Metropolis Monte Carlo algorithm is applied to an Ising ferromagnet with mixed spins of S=1/2 and S=1 and a crystal-field interaction. The critical temperatures and exponents both in the absence and presence of the crystal-field interaction are investigated with finite-size scaling theory. The critical temperature that we obtained in zero crystal field is very close to the high-temperature series-expansion result. The finite-size scaling results for the critical exponents yield the two-dimensional Ising-model values, which are in good agreement with those suggested by the universality hypothesis. The phase diagram of this model is obtained and comparisons are made with other approximate methods when available.
TL;DR: In this paper, a dynamic Monte Carlo renormalization group method is successfully applied to the equilibrium properties of the three-dimensional Ising model, and the critical exponent of the correlation length ν and critical amplitude of the surface tension σ 0 are estimated to be 0.6250 ± 0.0025 and 1.42 ± 0.04, respectively.
Abstract: From the non-equilibrium critical relaxation study of the two-dimensional Ising model, the dynamical critical exponent z is estimated to be 2.165 ± 0.010 for this model. The relaxation in the ordered phase of this model is consistent with exp (−√ t / τ ) behavior. The interface energy of the three-dimensional Ising model is studied and the critical exponent of the correlation length ν and the critical amplitude of the surface tension σ 0 are estimated to be 0.6250 ± 0.0025 and 1.42 ± 0.04, respectively. A dynamic Monte Carlo renormalization group method is successfully applied to the equilibrium properties of the three-dimensional Ising model.
TL;DR: An effective-field theory that has recently been used for studying higher-spin Ising models is extended to the transverse Ising model with an arbitrary spin S and the general formulation for evaluating the transition line in the Ω-T space and relevant statistical-mechanical quantities is derived.
Abstract: An effective-field theory that has recently been used for studying higher-spin Ising models is herein extended to the transverse Ising model with an arbitrary spin S. The general formulation for evaluating the transition line in the \ensuremath{\Omega}-T space and relevant statistical-mechanical quantities is derived. Numerical results are performed and analyzed for the particular cases S=3/2 and S=2.
TL;DR: In this article, a direct link between massive Ising model and arbitrary massive N = 2 supersymmetric QFT's in two dimensions is established, which explains why the equations which appear in the computation of spin-correlations in the non-critical Ising Model are the same as those describing the geometry of vacua in N=2 theories.
Abstract: We establish a direct link between massive Ising model and arbitrary massiveN=2 supersymmetric QFT's in two dimensions. This explains why the equations which appear in the computation of spin-correlations in the non-critical Ising model are the same as those describing the geometry of vacua inN=2 theories. The tau-function appearing in the Ising model (i.e., the spin correlation function) is reinterpreted in theN=2 context as a new “index”. In special cases this new index is related to the Ray-Singer analytic torsion, and can be viewed as a generalization of that to the loop space of Kahler manifolds.
TL;DR: In this paper, a closed set of Schwinger-Dyson equations for the two-matrix model in the large N limit was solved for correlation functions involving angular degrees of freedom.
TL;DR: In this article, the authors considered the quantum spin-1/2 Ising chain in a uniform transverse magnetic field, with an aperiodic sequence of ferromagnetic exchange couplings.
Abstract: We consider the quantum spin-1/2 Ising chain in a uniform transverse magnetic field, with an aperiodic sequence of ferromagnetic exchange couplings. This system is a limiting anisotropic case of the classical two-dimensional Ising model with an arbitrary layered modulation. Its formal solution via a Jordan-Wigner transformation enables us to obtain a detailed description of the influence of the aperiodic modulation on the singularity of the ground-state energy at the critical point. The key concept is that of thefluctuation of the sums of any number of consecutive couplings at the critical point. When the fluctuation isbounded, the model belongs to the “Onsager universality class” of the uniform chain. The amplitude of the logarithmic divergence in the specific heat is proportional to the velocity of the fermionic excitations, for which we give explicit expressions in most cases of interest, including the periodic and quasiperiodic cases, the Thue-Morse chain, and the random dimer model. When the couplings exhibit anunbounded fluctuation, the critical singularity is shown to be generically similar to that of the disordered chain: the ground-state energy has finite derivatives of all orders at the critical point, and an exponentially small singular part, for which we give a quantitative estimate. In themarginal case of a logarithmically divergent fluctuation, e.g., for the period-doubling sequence or the circle sequence, there is a negative specific heat exponentα, which varies continuously with the strength of the aperiodic modulation.
TL;DR: By a loop expansion around Parisi's mean field theory for an Ising spin-glass, it was shown that the overlap of the magnetization patterns belonging to two different temperatures, T and T', vanishes to any order, while the correlation overlap (sisj)T(sisj), calculated to first loop order (and, for technical reasons, for dimensions d>8 only) is found to decay exponentially, with a characteristic length approximately (T-T')-1 as mentioned in this paper.
Abstract: By a loop-expansion around Parisi's mean-field theory for an Ising spin-glass it is shown that the overlap of the magnetization patterns belonging to two different temperatures, T and T', vanishes to any order, (si)T(si)T=0, while the correlation overlap (sisj)T(sisj)T calculated to first loop order (and, for technical reasons, for dimensions d>8 only) is found to decay exponentially, with a characteristic length approximately (T-T')-1.
TL;DR: This ensemble where the partition function is simulated with a term in the action containing a varying magnetic field demonstrates on lattices with periodic boundary conditions that it is possible to enhance the appearance of order-order interfaces by many orders of magnitude.
Abstract: In analogy with a recently proposed multicanonical ensemble we introduce an ensemble where the partition function is simulated with a term in the action containing a varying magnetic field. Using this ensemble we demonstrate on lattices with periodic boundary conditions that it is possible to enhance the appearance of order-order interfaces by many orders of magnitude. To perform a stringent test of the method we consider the D=2 Ising model at \ensuremath{\beta}=0.5 and simulate square lattices up to size 100\ifmmode\times\else\texttimes\fi{}100. By a finite-size scaling analysis, the order-order interface tension per unit area is obtained. Our best infinite-volume extrapolation is in excellent agreement with Onsager's exact result.
TL;DR: In this paper, the low temperature dynamics of the Ising spin-glass in zero field with a discrete bond distribution was investigated via MC simulations and it was shown that the remanent magnetization decays algebraically and the temperature dependent exponents agree very well with the experimentally determined values.
Abstract: The low temperature dynamics of the three-dimensional Ising spin-glass in zero field with a discrete bond distribution is investigated via MC simulations. The thermoremanent magnetization is found to decay algebraically and the temperature dependent exponents agree very well with the experimentally determined values. The nonequilibrium autocorrelation function C(t, tw) shows a crossover at the waiting (or aging) time tw from algebraic quasiequilibrium decay for times t >tw with an exponent similar to one for the remanent magnetization.
TL;DR: In this article, the Ising antiferromagnet on a square lattice has a unique Gibbs measure if β(4−|h⥻) < 1/2ln(Pc/(1−Pc)), whereh denotes the external magnetic field, β the inverse temperature, andPc the critical probability for site percolation on that lattice.
Abstract: A uniqueness condition for Gibbs measures is given. This condition is stated in terms of (absence of) a certain type of percolation involving two independent realisations. This result can be applied in certain concrete situations by comparison with “ordinary” percolation. In this way we prove that the Ising antiferromagnet on a square lattice has a unique Gibbs measure if β(4−|h⥻)<1/2ln(Pc/(1−Pc)), whereh denotes the external magnetic field, β the inverse temperature, andPc the critical probability for site percolation on that lattice. SincePc is larger than 1/2, this extends a result by Dobrushin, Kolafa and Shlosman (whose proof was computer-assisted).
TL;DR: This model of M-component quantum rotors coupled by Gaussian-distributed random, infinite-range exchange interactions suggests that the critical properties of the transverse-field Ising model (believed to be identical to the M→1 limit) are the same as those of the M=∞ quantum rotor.
Abstract: We examine a model of M-component quantum rotors coupled by Gaussian-distributed random, infinite-range exchange interactions. A complete solution is obtained at M=\ensuremath{\infty} in the spin-glass and quantum-disordered phases. The quantum phase transition separating them is found to possess logarithmic violations of scaling, with no further modifications to the leading critical behavior at any order in 1/M; this suggests that the critical properties of the transverse-field Ising model (believed to be identical to the M\ensuremath{\rightarrow}1 limit) are the same as those of the M=\ensuremath{\infty} quantum rotors.
TL;DR: The phase transitions and critical properties of two types of inhomogeneous systems are reviewed in this article, where the authors combine mean field theory, scaling considerations, conformal transformations and perturbation theory.
Abstract: The phase transitions and critical properties of two types of inhomogeneous systems are reviewed. In one case, the local critical behaviour results from the particular shape of the system. Here scale-invariant forms like wedges or cones are considered as well as general parabolic shapes. In the other case the system contains defects, either narrow ones in the form of lines or stars, or extended ones where the couplings deviate from their bulk values according to power laws. In each case the perturbation may be irrelevant, marginal or relevant. In the marginal case one finds local exponents which depend on a parameter. In the relevant case unusual stretched exponential behaviour and/or local first-order transitions appear. The discussion combines mean field theory, scaling considerations, conformal transformations and perturbation theory. A number of examples are Ising models for which exact results can be obtained. Some walks and polymer problems are considered, too.
TL;DR: In this article, the authors considered layered systems with aperiodically modulated couplings and studied the effect of inhomogeneity on the critical behaviour, using scaling arguments a relevance/irrelevance-type criterion is formulated and for relevant inhomogeneities the scaling form of singular quantities is determined.
Abstract: The author considers layered systems with aperiodically modulated couplings and study the effect of inhomogeneity on the critical behaviour. Using scaling arguments a relevance/irrelevance-type criterion is formulated and for relevant inhomogeneities the scaling form of singular quantities is determined. The author has performed exact calculations for the surface magnetization of aperiodic quantum Ising chains and for a (1+1)-dimensional directed polymer in a longitudinal aperiodic environment. The results for both systems are in accord with scaling considerations. For a marginal aperiodicity the critical exponents are found to be non-universal.
TL;DR: In this article, the authors compute properties of the interface of the 3D Ising model for a wide range of temperatures, covering the whole region from the low temperature domain through the roughening transition to the bulk critical point.
Abstract: We compute properties of the interface of the 3-dimensional Ising model for a wide range of temperatures, covering the whole region from the low temperature domain through the roughening transition to the bulk critical point. The interface tension σ is obtained by integrating the surface energy density over the inverse temperature β. We use lattices of size L × L × T , with L up to 64, and T up to 27. The simulations with antiperiodic boundary conditions in T -direction are done with the Hasenbusch-Meyer interface cluster algorithm, which turns out to be very efficient. We demonstrate that in the rough phase the large distance behavior of the interface is well described by a massless Gaussian dynamics. The surface stiffness coefficient ϰ is determined. We also attempt to determine the correlation length ξ and study universal quantities like ξ 2 σ and ξ 2 ϰ. Results for the interfacial width on lattices up to 512 × 512 × 27 are also presented.