Showing papers on "Ising model published in 1988"

Finite size scaling analysis of Ising model block distribution functions

Abstract: The distribution function P L ( s ) of the local order parameters in finite blocks of linear dimension L is studied for Ising lattices of dimensionality d = 2,3 and 4. Apart from the case where the block is a subsystem of an infinite lattice, also the distribution in finite systems with free [ P L ( f ) ( s )] and periodic [ P L ( p ) ( s )] boundary conditions is treated. Above the critical point T c , these distributions tend for large L towards the same gaussian distribution centered around zero block magnetization, while below T c these distributions tend towards two gaussians centered at ± M , where M is the spontaneous magnetization appearing in the infinite systems. However, below T c the wings of the distribution at small | s | are distinctly nongaussian, reflecting two-phase coexistence. Hence the distribution functions can be used to obtain the interface tension between ordered phases. At criticality, the distribution functions tend for large L towards scaled universal forms, though dependent on the boundary conditions. These scaling functions are estimated from Monte Carlo simulations. For subsystem-blocks, good agreement with previous renormalization group work of Bruce is obtained. As an application, it is shown that Monte Carlo studies of critical phenomena can be improved in several ways using these distribution functions: ( i ) standard estimates of order parameter, susceptibility, interface tension are improved ( ii ) T c can be estimated independent of critical exponent estimates ( iii ) A Monte Carlo “renormalization group” similar to Nightingale's phenomenological renormalization is proposed, which yields fairly accurate exponent estimates with rather moderate effort ( iv ) Information on coarse-grained hamiltonians can be gained, which is particularly interesting if the method is extended to more general Hamiltonians.

Statistical Field Theory

Abstract: * Classical Equilibrium Statistical Mechanics * Magnetic Systems * The Ising Model * The Low-Temperature and High-Temperature Expansions * The Landau-Ginsbourg Model * Near the Transition * The Renormalization Group * Perturbative Evaluation of the Critical Exponents * Near Four Dimensions * On Spontaneous Symmetry Breaking * Other Models * The Transfer Matrix * Path Integrals for Quantum Mechanics * Semiclassical Methods * Relativistic Quantum Field Theory * Particle-Field Duality * Time-Dependent Correlations * The Approach to Equilibrium * The Stochastic Approach * Computer Simulation

Applied Conformal Field Theory

Abstract: These lectures consisted of an elementary introduction to conformal field theory, with some applications to statistical mechanical systems, and fewer to string theory. Contents: 1. Conformal theories in d dimensions 2. Conformal theories in 2 dimensions 3. The central charge and the Virasoro algebra 4. Kac determinant and unitarity 5. Identication of m = 3 with the critical Ising model 6. Free bosons and fermions 7. Free fermions on a torus 8. Free bosons on a torus 9. Affine Kac-Moody algebras and coset constructions 10. Advanced applications

Discontinuity of the magnetization in one-dimensional 1/¦x−y¦2 Ising and Potts models

Abstract: Results from percolation theory are used to study phase transitions in one-dimensional Ising andq-state Potts models with couplings of the asymptotic formJ x,y≈ const/¦x−y¦2. For translation-invariant systems with well-defined lim x→∞ x 2 J x =J + (possibly 0 or ∞) we establish: (1) There is no long-range order at inverse temperaturesβ withβJ +⩽1. (2) IfβJ +>q, then by sufficiently increasingJ 1 the spontaneous magnetizationM is made positive. (3) In models with 0

Heisenberg-to-Ising crossover in a random-field model with uniaxial anisotropy

Abstract: Uniaxial anisotropy causes a crossover between Ising and Heisenberg behavior in a random-field model. At random-field strengths and dimensionalities, intermediate between those which give either the Ising and Heisenberg domain states, a new ``boundary'' domain state appears. A novel criterion is also derived for domain-boundary roughening in the Heisenberg limit. The two-dimensional version of this model can be applied to ferromagnetic-antiferromagnetic sandwiches with interface randomness and leads to a prediction of an exchange anisotropy effect below a critical thickness of order the domain-wall width, and an enhancement of this effect below a second critical thickness.

Phase transition in the 3d random field Ising model

Abstract: We show that the three-dimensional Ising model coupled to a small random magnetic field is ordered at low temperatures. This means that the lower critical dimension,dl for the theory isdl≦2, settling a long controversy on the subject. Our proof is based on an exact Renormalization Group (RG) analysis of the system. This analysis is carried out in the domain wall representation of the system and it is inspired by the scaling arguments of Imry and Ma. The RG acts in the space of Ising models and in the space of random field distributions, driving the former to zero temperature and the latter to zero variance.

Monte Carlo study of growth in the two-dimensional spin-exchange kinetic Ising model

Abstract: Results obtained from extensive Monte Carlo simulations of domain growth in the two-dimensional spin-exchange kinetic Ising model with equal numbers of up and down spins are presented. Using different measures of domain size---including the pair-correlation function, the energy, and circularly-averaged structure factor---the domain size is determined (at T=0.5${T}_{c}$) as a function of time for times up to ${10}^{6}$ Monte Carlo steps. The growth law R(t)=A+${\mathrm{Bt}}^{1/3}$ is found to provide an excellent fit (within 0.3%) to the data, thus indicating that at long times the classical value of (1/3 for the exponent is correct. It is pointed out that this growth law is equivalent to an effective exponent for all times (as given by Huse) ${n}_{\mathrm{eff}}$(t)=(1/3-1)/3 C/R(t). No evidence for logarithmic behavior is seen. The self-averaging properties of the various measures of domain size and the variation of the constants A and B with temperature are also discussed. In addition, the scaling of the structure factor and anisotropy effects due to the lattice are examined.

Random‐field ising systems: A survey of current theoretical views

Abstract: Ising or Ising-like models in random fields are good representations of a large number of impure materials. The main attempts of theoretical treatments of these models--as far as they are relevant from an experimental point of view--are reviewed. A domain argument invented by Imry and Ma shows that the long-range order is not destroyed by weak random-fields in more than D = 2 dimensions. This result is supported by considerations of the roughening of an isolated domain wall in such systems: domain walls turn out to be well defined objects for D > 2, but arbitrarily convoluted for D < 2. Different approaches for calculating the roughness exponent ζ yield ζ= (5 - D)/3 in random-field systems. The application of ζ in incommensurate-commensurate critical behaviour is discussed. Non-classical critical behaviour occurs in random-field systems below D = 6 dimensions which is determined in general by three independent exponents. The new exponent yJ = θ= D/2 - σ corresponds to random-field renormalization...

THERMODYNAMICS OF OXYGEN ORDERING IN YBa2Cu3Oz

Abstract: It is shown that the (three-dimensional) tetragonal to orthorhombic transition in the 1-2-3 superconducting compound can be mapped onto a two-dimensional Ising model with anisotropic second neighbor effective pair interactions The Cluster Variation Method (CVM) has been used to calculate a phase diagram based on interaction parameters provided by first principles linear muffin-tin orbital-atomic sphere approximation (LMTO-ASA) calculations performed by P Sterne at Lawrence Livermore National Laboratory At high temperature, agreement with experimental phase transition data is excellent At low temperatures, two-dimensional ordering gives way to quasi one-dimensional states of order as described by the linear chain Ising model for which the thermodynamics are known exactly Structures produced by Monte Carlo simulation and theoretical diffraction patterns are in striking agreement with recent experimental results

Two-particle excitations in antiferromagnetic insulators.

Abstract: The energy of one and two holes in a Hubbard antiferromagnet for nonzero exchange, J, in the Ising limit is calculated within the Brinkman-Rice approximation. Only the p- and d-symmetry states bind with an energy of order J. The implications for superconductivity and antiferromagnetism of doped Hubbard insulators is discussed.

Dynamics of spin systems with randomly asymmetric bonds: Ising spins and Glauber dynamics.

Abstract: The stochastic dynamics of randomly asymmetric fully connected Ising systems is studied. We solve analytically the particularly simple case of fully asymmetric systems. We calculate the relaxation time of the autocorrelation function and show that the system remains paramagnetic even at zero temperature (T=0). The ferromagnetic phase is only slightly affected by the asymmetry, and the paramagnetic-to-ferromagnetic phase transition is characterized by a critical slowing down similar to second-order transition in symmetric (fully connected) systems. Monte Carlo simulations of a fully connected Ising system with random asymmetric interactions, both at finite and zero temperature, are presented. For finite T the autocorrelation function decays completely to zero for all strengths of the asymmetry. The T=0 behavior is more complex. In the fully asymmetric case the system is ergodic, with decaying autocorrelations, in agreement with the theoretical predictions. In the partially asymmetric case all flows terminate at fixed points (i.e., states which are stable to single spin flips). However, the typical time that it takes to converge to a fixed point grows exponentially with the size of the system. This convergence time varies from sample to sample and has a log-normal distribution in large systems. On time scales which are smaller than the convergence time, the system behaves ``ergodically,'' and the autocorrelation function decays to zero, much like the finite-temperature case.

Segregation and ordering at surfaces of transition metal alloys: the tight-binding Ising model

Abstract: From the electronic structure of the disordered alloy we derive an effective Ising Hamiltonian for segregation and ordering processes at transition metal alloy surfaces. In this tight-binding Ising model (TBIM), the Hamiltonian contains a linear term, quasi-concentration-independent, which proves to be very close to the difference in surface tensions between pure constituents and a quadratic one involving effective pair interactions larger at the surface than in the bulk. The former explains the success of the popular phenomenological approaches based on surface tension arguments and the latter could be of prime importance in surface ordering processes.

The Theory and Application of Axial Ising Models

Abstract: Publisher Summary The chapter aims for the following: first, to survey theoretical work on discrete spin models where competition in the Hamiltonian results in modulated ordering, and second, to discuss the relevance of such models to experimental systems Two compounds typical of those that will be considered are cerium antimonide and silicon carbide Cerium antimonide is an Ising ferromagnet, which locks into a large number of different modulated magnetic phases separated by first-order phase transitions In silicon carbide, on the other hand, the modulation is structural; the close-packed stacking sequence of the component layers can form long period patterns The canonical spin model that shows similar behavior is the axial next nearest- neighbor Ising, or ANNNI, model The phase diagram of the ANNNI model contains series of commensurate and incommensurate modulated phases of arbitrarily long wavelength These are stabilized by entropic effects, which dominate because of competition between short-range interactions The chapter presents a discussion on the chiral clock model, in which a different mechanism, chiral interactions, provides the competition, which leads to modulated phases The novel interface properties of this model are discussed

An intermediate phase with slow decay of correlations in one dimensional 1/|x−y|2 percolation, Ising and Potts models

Abstract: We rigorously establish the existence of an intermediate ordered phase in one-dimensional 1/|x−y|2 percolation, Ising and Potts models. The Ising model truncated two-point function has a power law decay exponent θ which ranges from its low (and high) temperature value of two down to zero as the inverse temperature and nearest neighbor coupling vary. Similar results are obtained for percolation and Potts models.

Lattice field theory as a percolation process.

Abstract: For a given lattice spin or gauge theory an associated correlated bond or plaquette percolation process is constructed. It is conjectured to reproduce the universal scaling behavior of the original model. Different field theories lead to different cluster weights generalizing a result by Fortuin and Kasteleyn for Potts models. The new representation lends itself to the design of Monte Carlo algorithms with reduced critical slowing down.

General cluster updating method for Monte Carlo simulations.

Abstract: A general cluster updating algorithm for Monte Carlo simulations is presented. The method contains arbitrary probability functions which can be used to minimize the relaxation time. It is applicable to systems where the interaction energy has a global (discrete or continuous) symmetry. Special cases cover standard site-by-site updating as well as the cluster updating method proposed by Swendsen and Wang for Ising and Potts models.

Thermal fluctuations in some random field models

Abstract: Exact identities are derived for a family of models including (a) a domain wall in a random field Ising model (RFIM), and (b) the random anisotropyXY model in the no-vortex approximation. In particular, the second moment of thermal fluctuations is not affected by frozen randomness. It is checked in a one-dimensional model that higher moments are on the contrary strongly enhanced. Thus, thermal fluctuations are strongly non-Gaussian. This reflects excursions between remote potential wells in the phase space. It is shown exactly that the Imry-Ma argument yields a correct evaluation of the field-induced fluctuations for the one-dimensional model.

Free energy of the solvable chiral Potts model

Abstract: Very recently, it has been shown that there are chiralN-state Potts models in statistical mechanics that satisfy the star-triangle relation. Here it is shown that the relation implies that the free energy (and its derivatives) satisfies certain functional relations. These can be used to obtain the free energy: in particular, we expand about the critical case and find that the exponent α is 1−2/N.

Critical exponents for long-range interactions

Abstract: Long-range components of the interaction in statistical mechanical systems may affect the critical behavior, raising the system's ‘effective dimension’. Presented here are explicit implications to this effect of a collection of rigorous results on the critical exponents in ferromagnetic models with one-component Ising (and more genrally Griffiths=Simon class) spin variables. In particular, it is established that even in dimensions d<4 if a ferromagnetic Ising spin model has a reflection-positive pair interaction with a sufficiently slow decay, e.g. as J x=1/|x| d+δ with 0<δ≤d/2, then the exponents $$\hat \beta $$ , δ, γ and Δ4 exist and take their mean-field values. This proves rigorously an early renormalization-group prediction of Fisher, Ma and Nickel. In the converse direction: when the decay is by a similar power law with δ>-2, then the long-range part of the interaction has no effect on the existent critical exponent bounds, which coincide then with those obtained for short-range models.

Observation of 3D-Ising exponents in micellar solutions.

Abstract: Mesure de la susceptibilite osmotique du melange eau/n-dodecyl octaoxyethylene glycol monoether

Exactly solvable Potts models, bond- and tree-like percolation on dynamical (random) planar lattice

Abstract: It is shown that q-state Potts models on dynamical planar (random) lattice (DPL) are exactly solvable: the problem is reduced to some large N saddle point equations. Further simplications occur in the cases of the Ising model on DPL (q = 2, solved in [1], of bond percolation on random quenced Lattice (q → 1) and of tree-like percolation on DPL (q → 0). For the last one the exact free energy is calculated and the existence of phase transition is shown.

The nature of attractors in an asymmetric spin glass with deterministic dynamics

Abstract: The authors study the attractors in an infinite-range Ising spin-glass model with deterministic dynamics where the interactions have asymmetry, specified by a parameter k. They find a duality relation between the attractors for models with asymmetry parameters k and 1/k. The attractors are fixed points or limit cycles of short length, except for k=1, at which the average cycle length diverges, reminiscent of a phase transition, and the model has many similarities to the random map model as well as to the infinite-range symmetric spin glass in thermal equilibrium, including the fact that a few attractors dominate the weight. The extent of this dominance varies from sample to sample and so is given by a non-trivial probability distribution, Pi (Y), which they compute numerically.

Models in Statistical Physics and Quantum Field Theory

Abstract: 1. Introduction.- 1.1 Phase Transitions - Critical Phenomena.- 1.1.1 Historical Survey.- 1.1.2 Gas-Liquid Transition.- 1.1.3 Ferromagnetism.- 1.1.4 Critical Exponents.- 2. Spin Systems.- 2.1 Ising Model - General Results.- 2.1.1 Introduction.- 2.1.2 Ising Model in One Dimension.- 2.1.3 Duality.- 2.1.4 Peierls' Argument.- 2.1.5 Correlation Inequalities.- 2.2 Heisenberg Model.- 2.2.1 Bogoliubov Inequality.- 2.2.2 Absence of Spontaneous Magnetization for d = 1 and d = 2.- 2.2.3 Existence of a Phase Transition for d Greater than or Equal to Three.- 2.3 o4-Model.- 2.3.1 Random Walk on a Lattice.- 2.3.2 Polymer Representation.- 2.3.3 Correlation Inequality.- 2.3.4 Continuum Limit.- 2.4 Two-Dimensional Ising Model.- 2.4.1 Transfer Matrix.- 2.4.2 Klein-Jordan-Wigner Transformation.- 2.4.3 Fourier Transformation.- 2.4.4 Bogoliubov Transformation.- 3. Two-Dimensional Field Theory.- 3.1 Solitons.- 3.1.1 Inverse Scattering Formalism.- 3.1.2 Solving Certain Nonlinear Partial Differential Equations.- 3.1.3 A Model for Polyacetylene.- 3.2 Sectors in Field Theoretical Models.- 3.2.1 External Field Problems.- 3.2.2 The Schwinger Model.- 4. Lattice Gauge Models.- 4.1 Formulation.- 4.1.1 Axioms.- 4.1.2 The Rolling Ball.- 4.1.3 Classical Field Theory.- 4.1.4 Formulation of Lattice Gauge Models.- 4.1.5 Fermions on the Lattice.- 4.2 Rigorous Results.- 4.2.1 Faddeev-Popov "Trick" on a Lattice.- 4.2.2 Physical Positivity = Osterwalder-Schrader Positivity.- 4.2.3 Cluster Expansion.- 4.2.4 Confinement.- 4.2.5 Remarks on Numerical Methods.- 4.2.6 Recent Developments.- 5. String Models.- 5.1 Introduction to Strings.- 5.1.1 Classical Mechanics of Strings.- 5.1.2 Quantization of the Bosonic String.- 5.1.3 Fermionic Strings and Superstrings.- 6. Renormalization Group.- 6.1 Formulation.- 6.1.1 Scaling Laws.- 6.1.2 Kadanoff's Block Spin Method.- 6.1.3 Wilson's Renormalization Group Ideas.- 6.1.4 Ising Model d = 1.- 6.2 Application of the Renormalization Group Ideas to Special Models.- 6.2.1 Central Limit Theorem.- 6.2.2 Hierarchical Model.- 6.2.3 Two-Dimensional Ising Model.- 6.2.4 Ginzburg-Landau-Wilson Model.- 6.2.5 Feigenbaum's Route to Chaos.- General References.

Universal configurational structure in two-dimensional scalar models

Abstract: Monte Carlo methods are used to explore the universal configurational structure of two-dimensional spin- 1/2, spin-1 and border- phi 4 models. Comparison of spin- 1/2 and spin-1 data provides evidence that the magnetisation distribution (effectively the Helmholtz free-energy function) and its coupling derivative (effectively the internal-energy function) constitute readily accessible signatures of a universality class. It is shown that, when allowance is made for relatively large corrections-to-scaling effects, the behaviour of the border- phi 4 model may be satisfactorily matched to that of the other two models, substantiating the view that the border model does indeed belong to the Ising universality class.

Exact solutions for Ising-model even-number correlations on planar lattices.

Abstract: A systematic and unifying method is developed and demonstrated for obtaining exact solutions of n-site (n even integer) Ising correlations on various planar lattices. The scheme, which is exceedingly more simple than using solely traditional Pfaffian techniques, embodies five mapping theorems in alliance with algebraic correlation identities. In the theoretical framework, the triangular Ising model plays an overarching role. In particular, considering a select 7-site cluster of the triangular Ising model, the knowledge of all its 11 even-number correlations defined upon this cluster (where only four of the correlations need to be actually calculated by Pfaffian procedures) is shown to be sufficient for determining exactly all honeycomb, decorated-honeycomb, and kagom\'e Ising model even-number correlations upon their correspondingly select 10-site, 19-site, and 9-site clusters, respectively. The relative ease and direct applicability of the present approach are highlighted not only by the resulting large numbers of n-site (n even-integer) correlation solutions (e.g., approximately 85 and 50 for the honeycomb and kagom\'e Ising models, respectively) and their large ${n}_{\mathrm{max}}$ values (${n}_{\mathrm{max}=8}$,10,18 for the kagom\'e, honeycomb, and decorated-honeycomb Ising models, respectively) but also by the realization that the exact solutions for Ising multisite correlations upon the kagom\'e lattice (one of the four regular lattices in two dimensions) are apparently the first to explicitly appear in the literature beyond its nearest-neighbor pair correlation (energy).

Chemical and elastic effects on isostructural phase diagrams: The ɛ - G approach

Abstract: Numerous theoretical models of temperature-composition phase diagrams of isostructural binary alloys are based on the configurational Ising Hamiltonian in which the many-body configurational interaction energies ${\ensuremath{\varepsilon}}^{(n)}$ are taken as (volume-independent) constants (the ``\ensuremath{\varepsilon}-only'' approach). Other approaches postulate phenomenologically composition-dependent but configuration- (\ensuremath{\sigma}-) independent elastic energies. We show that under the commonly encountered situation where molar volumes at fixed composition (x) do not depend on the state of order, a new approach is pertinent: We prove that the physically relevant Hamiltonian (the ``\ensuremath{\varepsilon}-G approach'') includes the configuration-dependent (but concentration-independent) ``chemical'' interaction energies ${\ensuremath{\varepsilon}}^{(n)}$, plus a composition-dependent (but configuration-independent) elastic energy G(x). We compute the elastic term G(x) from the structural and elastic properties of ordered intermetallic systems. We show that inclusion of G(x) into the conventional configurational (\ensuremath{\varepsilon}-only) Hamiltonian cures many of the shortcomings of such Ising models in describing actual alloy phase diagrams. In particular, addition of the elastic energy G(x) leads to the following features: (i) narrower single-phase regions and broader mixed-phase regions, (ii) shift of the triple point to substantially higher temperatures, (iii) the mixing enthalpies of the disordered phases become much closer to the experimental data, and (iv) the possibility of the occurrence of metastable long-range-ordered compounds inside the miscibility gap. Cluster-variation and Monte Carlo calculations on model Hamiltonians and on the Cu-Au system are used to illustrate these points.

Phase transition of the mixed spin system with a random crystal field

Abstract: The effect of a random crystal field on the transition temperature of the spin- 1 2 and spin-1 mixed Ising spin system is investigated with the use of an effective-field theory with correlations. We find a variety of interesting phenomena resulting from the fluctuation of the crystal field interaction.

Phase diagram of a lattice microemulsion model

Abstract: We derive the phase diagram of a recently proposed model of microemulsions. The model is equivalent to an Ising model with nearest‐neighbor interaction parameter J, diagonal‐neighbor interaction parameter 2M, and further‐neighbor interaction parameter M. We find the regions of stability of the various phases in the j(=J/kT), m(=M/kT) plane, both by solution of the mean‐field equations at finite j and m and by an exact analysis at T=0(m→−∞, j→±∞). The disordered (paramagnetic) region is bounded by an ellipse and two of its tangents. We concentrate our attention on the phases of periodic structure near the paramagnetic boundary, on the low‐temperature phases, and on the connections between them.