Showing papers on "Ising model published in 1991"

Nature and dynamics of the spin-state interconversion in metal complexes

Abstract: Spin-state transitions in metal complexes which are driven by a change of temperature T or pressure p are always associated with a considerable reorganization of molecular geometry. The change involves metal-ligand bond lengths R, bond angles, and a variation of ligand orientation. In particular, the elongation 4R by up to ∼ 10% occurring in the course of the LS → HS conversion produces an expansion of molecular volume ΔV ≌ 25 A3 per metal atom. The average crystal structure for given values of T and p is reproduced by the fractional occupancy of the individual structures of the high-spin (HS) and low-spin (LS) isomer. The transitions are reasonably well described by a number of theoretical models which are equivalent to the Bragg and Williams approximation of the Ising model. The dynamics of the spin-state transitions in solution, based on measurements by ultrasonic and photo-perturbation techniques, is in general rapid with rate constants between 4 × 105 and 3 × 108 s−1. Similar results are obtained for the spin conversion in solid complexes where the line shape analysis of Mossbauer spectra based on the theory of Blume and Tjon is applied. The dynamics may be rationalized employing one-dimensional cross sections through Gibbs free-energy surfaces G = G(R), an alternative being the comparison of the results with quantum-mechanical calculations for a radiationless non-adiabatic multiphonon process.

A hierarchically constrained kinetic Ising model

Abstract: A concrete model for hierarchically constrained dynamics in the sense proposed by Palmer et al. (Phys. Rev. Lett.53, 958 (1984)) is presented. The model is a kinetic Ising chain with an asymmetric kinetic constraint, allowing a spin to flip only if its neighbour to the right is in the up spin state. The spin autocorrelation function is obtained by numerically exact calculation for finite chain length up toL=9 and by Monte Carlo simulation for effectively infinite chain length. The Kohlrausch-Williams-Watts formula is found to fit the results only with limited accuracy, and within limited time intervals. We also performed an analytical calculation using an effective-medium approximation. The approximation leads to a spurious blocking transition at a critical up spin concentrationc=0.5.

Multicritical phase diagrams of the Blume-Emery-Griffiths model with repulsive biquadratic coupling.

Abstract: Six new phase diagrams, including a novel multicritical topology and two new ordered phases, high-entropy ferrimagnetic and antiquadrupolar, are found in the spin-1 Ising model with only nearest-neighbor interactions, for negative biquadratic couplings. Thus, the global phase diagram of this simple spin system includes nine distinct topologies and three ordered phases. It is indicated that these results, obtained by mean-field theory, are applicable to three-dimensional systems.

Linked-cluster expansion around the atomic limit of the Hubbard model.

Abstract: We develop a perturbation expansion in the intersite hopping around the atomic limit of the Hubbard model. It is valid for arbitrary finite temperatures and interaction strengths. Diagrammatic rules that determine the grand-canonical potential and the Green's functions are derived. They reduce the calculation of any finite-order contribution to simple algebra. This opens the way for series extrapolations from computer-aided high-finite-order evaluations. Discrepancies in earlier expansions around the atomic limit are clarified. The present expansion scheme involves only connected diagrams with unrestricted lattice sums. This allows one to perform a vertex renormalization as for the linked-cluster expansion of the Ising model. The renormalized perturbation expansion can be used to construct self-consistent approximations which are automatically exact in the atomic limit. In the limit of high lattice dimensions, only fully two-particle reducible embeddings of diagrams on the lattice contribute. The single-particle properties of the infinite-dimensional Hubbard model reduce to those of independent tight-binding fermions hopping between dressed sites.

Critical droplets and metastability for a Glauber dynamics at very low temperatures

Abstract: We consider the metastable behavior in the so-called pathwise approach of a ferromagnetic spin system with a Glauber dynamics in a finite two dimensional torus under a positive magnetic field in the limit as the temperature goes to zero. First we consider the evolution starting from a single rectangular droplet of spins +1 in a sea of spins −1. We show that small droplets are likely to disappear while large droplets are likely to grow; the threshold between the two cases being sharply defined and depending only on the external field. This result is used to prove that starting from the configuration with all spins down (−1) the pattern of evolution leading to the more stable configuration with all spins up (+1) approaches, as the temperature vanishes, a metastable behavior: the system stays close to −1 for an unpredictable time until a critical square droplet of a precise size is eventually formed and nucleates the decay to +1 in a relatively short time. The asymptotic magnitude of the total decay time is shown to be related to the height of an energy barrier, as expected from heuristic and mean field studies of metastability.

Bose condensation in quasi-one-dimensional antiferromagnets in strong fields.

Abstract: Recent experiments show that axially symmetric integer-spin antiferromagnetic chains undergo a phase transition at a critical applied magnetic field It was argued, using Landau-Ginzburg theory, that this is one-dimensional Bose condensation The theory is further analyzed at the mean-field and Gaussian level Then some exact results concerning the critical behavior are determined using known results on one-dimensional Bose fluids These are shown to be consistent with recent numerical simulations on spin chains Breaking of axial symmetry produces crossover to a two-dimensional Ising transition

How to expand around mean-field theory using high-temperature expansions

Abstract: High-temperature expansions performed at a fixed-order parameter provide a simple and systematic way to derive and correct mean-field theories for statistical mechanical models. For models like spin glasses which have general couplings between spins, the authors show that these expansions generate the Thouless-Anderson-Palmer equations at low order. They explicitly calculate the corrections to TAP theory for these models. For ferromagnetic models, they show that their expansions can easily be converted into 1/d expansions around mean-field theory, where d is the number of spatial dimensions. Only a small finite number of graphs need to be calculated to generate each order in 1/d for thermodynamic quantities like free energy or magnetization. Unlike previous 1/d expansions, the expansions are valid in the low-temperature phases of the models considered. They consider alternative ways to expand around mean-field theory besides 1/d expansions. In contrast to the 1/d expansion for the critical temperature, which is presumably asymptotic, these schemes can be used to devise convergent expansions for the critical temperature. They also appear to give convergent series for thermodynamic quantities and critical exponents. They test the schemes using the spherical model, where their properties can be studied using exact expressions.

Strict monotonicity for critical points in percolation and ferromagnetic models

Abstract: When is the numerical value of the critical point changed by an enhancement of the process or of the interaction? Ferromagnetic spin models, independent percolation, and the contact process are known to be endowed with monotonicity properties in that certain enhancements are capable of shifting the corresponding phase transition in only an obvious direction, e. g., the addition of ferromagnetic couplings can only increase the transition temperature. The question explored here is whether enhancements do indeed change the value of the critical point. We present a generally applicable approach to this issue. For ferromagnetic Ising spin systems, with pair interactions of finite range ind⩾2 dimensions, it is shown that the critical temperatureTc is strictly monotone increasing in each coupling, with the first-order derivatives bounded by positive functions which are continuous on the set of fullyd-dimensional interactions. For independent percolation, with 0

From classical to quantum glass

Abstract: We study the effects of a transverse magnetic field on the dynamics of the randomly diluted, dipolar coupled, Ising magnet LiHo_(0.167)Y_(0.833)F_4. The transverse field mixes the eigenfunctions of the ground-state Ising doublet with the otherwise inaccessible excited-state levels. We observe a rapid decrease in the characteristic relaxation times, large changes in the spectral form of the relaxation, and a depression of the spin-glass transition temperature with the introduction of quantum fluctuations.

Correlation Functions of Integrable 2d Models of the Relativistic Field Theory:. Ising Model

Abstract: A program is proposed to study numerically the correlation functions in massive integrable 2D relativistic field theories. It relies crucially on the exact form factors of fields which can be reconstructed from the factorized scattering data. The correlation functions are expressible as infinite sums over intermediate asymptotic states. We suggest using computer power to perform the summation numerically. The convergence of the sum is tested for the simplest example of the scaling Ising spin-spin correlations (without magnetic field).

Truncated fermionic space approach to the critical 2-D Ising model with magnetic field

Abstract: The critical Ising model perturbed by the spin field (conjugated to the magnetic field) is studied numerically by the method of truncated fermionic space of states. The matrix elements of the perturbed Hamiltonian are found between the states of free Majorana fermions living on a finite-length circle. The Hamiltonian spectrum is studied numerically in the level-5 truncated space. Reasonable estimations are obtained for the vacuum energy, a part of the mass spectrum and the simplest scattering amplitude of the perturbed system.

Generalized Thue-Morse chains and their physical properties

Abstract: We study the physical properties of the Thue-Morse chain and its generalizations. After a preliminary discussion of its basic features (e.g., structure factor, location, and relative magnitude of spectral gaps), we focus on (1) the trace maps of generalized Thue-Morse lattices, (2) a detailed analysis of the attractor of the associated dynamical system, (3) the electronic spectra through the trace-map approach, (4) spin excitations in a quantum Ising model in a transverse magnetic field, (5) light transmission through a multilayer, and (6) the diamagnetic properties of Thue-Morse superconducting wire networks and Josephson-junction arrays.

Gibbs Random Fields: Cluster Expansions

Abstract: 1. Gibbs Fields (Basic Notions).- 0 First Acquaintance with Gibbs Fields.- 1 Gibbs Modifications.- 2 Gibbs Modifications under Boundary Conditions and Definition of Gibbs Fields by Means of Conditional Distributions.- 2. Semi-Invariants and Combinatorics.- 1 Semi-Invariants and Their Elementary Properties.- 2 Hermite-Ito-Wick Polynomials. Diagrams. Integration by Parts.- 3 Estimates on Moments and Semi-Invariants of Functional of Gaussian Families.- 4 Connectedness and Summation over Trees.- 5 Estimates on Intersection Number.- 6 Lattices and Computations of Their Moebius Functions.- 7 Estimate of Semi-Invariants of Partially Dependent Random Variables.- 8 Abstract Diagrams (Algebraic Approach).- 3. General Scheme of Cluster Expansion.- 1 Cluster Representation of Partition Functions and Ensembles of Subsets.- 2 Cluster Expansion of Correlation Functions.- 3 Limit Correlation Function and Cluster Expansion of Measures.- 4 Cluster Expansion and Asymptotics of Free Energy. Analyticity of Correlation Functions.- 5 Regions of Cluster Expansions for the Ising Model.- 6 Point Ensembles.- 4. Small Parameters in Interactions.- 1 Gibbs Modifications of Independent Fields with Bounded Potential.- 2 Unbounded Interactions in the Finite-Range Part of a Potential.- 3 Gibbs Modifications of d-Dependent Fields.- 4 Gibbs Point Field in Rv.- 5 Models with Continuous Time.- 6 Expansion of Semi-Invariants. Perturbation of a Gaussian Field.- 7 Perturbation of a Gaussian Field with Slow Decay of Correlations.- 8 Modifications of d-Markov Gaussian Fields (Interpolation of Inverse Covariance).- 5. Expansions Around Ground States (Low-Temperature Expansions).- 1 Discrete Spin: Countable Number of Ground States.- 2 Continuous Spin: Unique Ground State.- 3 Continuous Spin: Two Ground States.- 6. Decay of Correlations.- 1 Hierarchy of the Properties of Decay of Correlations.- 2 An Analytic Method of Estimation of Semi-Invariants of Bounded Quasi-Local Functionals.- 3 A Combinatorial Method of Estimation of Semi-Invariants in the Case of Exponentially-Regular Cluster Expansion.- 4 Slow (power) Decay of Correlations.- 5 Low-Temperature Region.- 6 Scaling Limit of a Random Field.- 7. Supplementary Topics and Applications.- 1 Gibbs Quasistates.- 2 Uniqueness of Gibbs Fields.- 3 Compactness of Gibbs Modifications.- 4 Gauge Field with Gauge Group Z2.- 5 Markov Processes with Local Interaction.- 6 Ensemble of External Contours.- Concluding Remarks.- Bibliographic Comments.- References.

Monte Carlo investigation of critical dynamics in the three-dimensional Ising model.

Abstract: We report the results of a Monte Carlo investigation of the (equilibrium) time-displaced correlation functions for the magnetization and energy of a simple cubic Ising model as a function of time, temperature, and lattice size. The simulations were carried out on a CDC CYBER 205 supercomputer employing a high-speed, vectorized multispin coding program and using a total of 5\ifmmode\times\else\texttimes\fi{}${10}^{12}$ Monte Carlo spin-flip trials. We used L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L lattices with periodic boundary conditions and L as large as 96. The short-time and long-time behaviors of the correlation functions are analyzed by fits to a sum of exponential decays, and the critical exponent z for the largest relaxation time is extracted using a finite-size-scaling analysis. Our estimate z=2.04\ifmmode\pm\else\textpm\fi{}0.03 resolves an intriguing contradiction in the literature; it satisfies the theoretical lower bound and is in agreement with the prediction obtained by \ensuremath{\epsilon} expansion. We also consider various small systematic errors that typically occur in the analysis of relaxation functions and show how they can lead to spurious results if sufficient care is not exercised.

Square-lattice Heisenberg antiferromagnet at T =0

Abstract: The spin-1/2 and spin-1 Heisenberg antiferromagnets on a square lattice are studied via series expansions around the Ising limit. Series are calculated for the ground-state energy, staggered magnetization, transverse susceptibility, staggered parallel susceptibility, and mass gap. Extrapolating these series to the isotropic limit, we find extremely good agreement with the predictions of spin-wave theory.

Phenomena due to strain coupling in phase transitions.

Abstract: A theoretical model of an order-disorder phase transition in a crystal has been developed with strain-induced coupling between Ising spins. Strain coupling leads to long-ranged renormalized interactions between the spins which result in ferroelastic-antiferroelastic phase transitions. Computer simulations show textured microstructures evolving for some cases, similar to tweed textures. In other cases the long correlation length of strain suppresses nucleation, and the rate of (dis)ordering follows mean-field theory, very different from «nucleation and growth»

The origin of glassy dynamics in the 2D facilitated kinetic Ising model

Abstract: Monte Carlo simulation data from the two‐dimensional facilitated kinetic Ising model proposed by Fredrickson and Andersen is examined with the goal of understanding the processes responsible for the characteristic features of glassy dynamics. The spatial distribution of spin flip rates is found to be highly nonuniform with pockets of rapidly relaxing spins surrounded by kinetically locked domains. The slow relaxation of these latter domains, which gives rise to the characteristic long time tail of the linear response function, is due to the action of rare clusters of spins which are able to propagate their influence throughout the sample. An analytic expression is derived for the density of these active sites which is found to fully account for the non‐Arrhenius temperature dependence of the relaxation time in this model. The consequences of these results for both theories and experiments in structural glasses are discussed.

Phase diagrams of Ising models on Husimi trees. I. Pure multisite interaction systems

Abstract: Lattice spin systems with multisite interactions have rich and interesting phase diagrams. We present some results for such systems involving Ising spins (σ=±1) using a generalization of the Bethe lattice approximation. First, we show that our approach yields good approximations for the phase diagrams of some recently studied multisite interaction systems. Second, a multisite interaction system with competing interactions is investigated and a strong connection with results from the theory of dynamical systems is made. We exhibit a full bifurcation diagram, chaos, period-3 windows, etc., for the magnetization of the base site of this system.

Ferromagnetism in small clusters.

Abstract: Magnetization of small ferromagnetic clusters at finite temperatures has been studied using the Ising model and Monte Carlo techniques. The magnetization of finite clusters is reduced from the bulk value, and increases with the external magnetic field and with the cluster size. The results explain qualitatively the recent observations by de Heer, Milani, and Chatelain of the reduction with decreasing cluster size of the average magnetic moment in small iron clusters.

Spontaneous alignment of frustrated bonds in an anisotropic, three-dimensional Ising model

Abstract: The Ising model on a three-dimensional cubic lattice with all plaquettes in the x-y frustrated plane is studied by use of a Monte Carlo technique; the exchange constants are of equal magnitude, but have varying signs. At zero temperature, the model has a finite entropy and no long-range order. The low-temperature phase is characterized by an order parameter measuring the ${\mathit{openZ}}_{4}$ symmetry of lattice rotations which is invariant under Mattis gauge transformation; fluctuations lead to the alignment of frustrated bonds into columns and a fourfold degeneracy. An additional factor-of-2 degeneracy is obtained from a global spin flip. The order vanishes at a critical temperature by a transition that appears to be in the universality class of the D=3, XY model. These results are consistent with the theoretical predictions of Blankschtein et al. This Ising model is related by duality to phenomenological models of two-dimensional frustrated quantum antiferromagnets.

Non-equilibrium dynamics of the Ising model for T less-than/equal-toTc

Abstract: The growth of order in Ising models with nonconserved order parameter is considered for quenches to final temperatures Tf=0 and Tf=Tc. The results of numerical simulations in spatial dimension d=2 are presented. In all cases a scaling regime is entered for sufficiently long times, where the characteristic length scale is the 'domain size' L(t) approximately t12/, for Tf=0, and the 'nonequilibrium correlation length', xi (t) approximately t1z/, for Tf=Tc. The equal-time correlation function has the expected scaling forms f(r/L(t)) and r-(d-2+ eta )fc(r/ xi (t)) for Tf=0 and Tc respectively. The scaling function fc(x) has interesting short-distance behaviour which is elucidated using scaling arguments and by in - and 1/n-expansions. The T=0 scaling function f(x) depends on whether the spin correlations present in the initial conditions are of long or short range, as does the exponent which describes the decay of the autocorrelation function, A(t)=((Si(t))Si(O)) approximately L(t)-. Results for a quench from the equilibrium critical state to Tf=0 are consistent with theoretical predictions.

Existence of a Sm-C Grain Boundary Phase at the Chiral MAC Point

Abstract: Based on a mean field analysis of the Chen-Lubensky model we extend an earlier discussion regarding the effect of chirality on the region of the phase diagram where the Cholesteric, Smectic-A, and Smectic- C* phases meet (i.e., the N* AC* point). We show that, on the Sm-C* side of the phase diagram, a second twist grain boundary (or chiral smectic) phase occurs. This phase, the “TGB C ”, is a highly dislocated version of Smectic-C. According to the deGennes analogy, this phase corresponds to Abri- kosov's flux lattice when the Ginzburg parameter K is negative. We also show that an Ising like transition between the TGB A and TGB C occurs and that this phase boundary, together with the Sm-A/TGB A Sm-C/TGBc, and the x-y like Sm-A/Sm-C* phase boundaries, meet at a multicritical point.