Showing papers on "Ising model published in 1980"

Order as an effect of disorder

Abstract: A generalized frustrated Ising model on a two-dimensional lattice is considered. This model is paramagnetic at zero temperature but ferromagnetic provided 0 < T < T c. The effect of dilution on this system is also investigated, and long range order is shown to be restored in the dilute model under certain conditions involving concentration, temperature and interactions which are discussed in comparison with usual percolation.

Duality in field theory and statistical systems

Abstract: This paper presents a pedagogical review of duality (in the sense of Kramers and Wannier) and its application to a wide range of field theories and statistical systems. Most of the article discusses systems in arbitrary dimensions with discrete or continuous Abelian symmetry. Globally and locally symmetric interactions are treated on an equal footing. For convenience, most of the theories are formulated on a $d$-dimensional (Euclidean) lattice, although duality transformations in the continuum are briefly described. Among the familiar theories considered are the Ising model, the $x\ensuremath{-}y$ model, the vector Potts model, and the Wilson lattice gauge theory with a ${Z}_{N}$ or $U(1)$ symmetry, all in various dimensions. These theories are all members of a more general heirarchy of theories with interactions which are distinguished by their geometrical character. For all these Abelian theories it is shown that the duality transformation maps the high-temperature (or, for a field theory, large coupling constant) region of the theory into the low-temperature (small coupling constant) region of the dual theory, and vice versa. The interpretation of the dual variables as disorder parameters is discussed. The formulation of the theories in terms of their topological excitations is presented, and the role of these excitations in determining the phase structure of the theories is explained. Among the other topics discussed are duality for the Abelian Higgs model and related models, duality transformations applied to random systems (such as theories of a spin glass), duality transformations in the "lattice Hamiltonian" formalism, and a description of attempts to construct duality transformations for theories with a non-Abelian symmetry, both on the lattice and in the continuum.

Ising model with solitons, phasons, and "the devil's staircase"

Abstract: We have analyzed the modulated phase of an Ising model with competing interactions in an effort to increase the understanding of the spatially modulated phases found in many physical systems. The analysis has three stages. First, the mean-field phase diagram is calculated numerically. A large, possibly infinite, number of phases where the periodicity of the ordered structure is commensurate with the lattice is found. The resulting periodicity-versus-temperature curve thus probably has an infinity of steps; i.e., it exhibits "the devil's staircase" behavior. Then the mean-field theory is analyzed analytically, and it is shown that the stability of the commensurate phases can be understood within a domain-wall or "soliton" theory. The solitons from a regular lattice near the transitions to the commensurate phases. The elementary excitations in the solition lattice are the phasons. Third, the effects of temperature-induced fluctuations, ignored in the mean-field theory, are estimated by calculating the entropy contribution to the free energy from the phasons. It is found that the stability ranges of the commensurate phases are reduced, but the staircase survives at finite temperatures. On the basis of our calculations a phase diagram is constructed.

Clusters and Ising critical droplets: a renormalisation group approach

Abstract: The Migdal-Kadanoff renormalisation group for two-dimensions is employed to obtain the global phase diagram for the site-bond correlated percolation problem. It is found that the Ising critical point (K=Kc,H=O) is a percolation point for a range of bond probability rho B such that 1>or= rho B>or=1-e-2Kc. In particular, as rho B approaches 1-e-2Kc, the percolation clusters become less compact and coincide with the Ising critical droplets.

Ordering in strongly fluctuating condensed matter systems

Abstract: Ordering in Strongly Fluctuating Systems: Introductory Comments.- 1. Introduction.- 2. A Theorist's Ideal Glass.- 3. Systems Far From Equilibrium.- Phase Transitions in Low-Dimensional Systems and Renormalization Group Theory.- 1. Phase Transitions and Some Simple Spin Models.- 2. Fluctuations and the Lower Critical Dimension.- 3. Values of Lower Critical Dimensionality for Some Spin Models.- 4. Fluctuations.- 5. Introduction to the Renormalization Group.- Real-Space Renormalization-Group Method for Quantum Systems.- Upper Marginal Dimensionality, Concept and Experiment.- 1. Phenomenological Description.- 2. Mean Field Theory.- 3. Ginzburg Criterion.- 4. Experiments on LiTbF4.- 5. Conclusion.- Lower Marginal Dimensionality. X-Ray Scattering from the Smectic-A Phase of Liquid Crystals.- 1. Introduction.- 2. The Nematic and Smectic A Phases of Liquid Crystals.- 3. The Correlation Function in the Harmonic Approximation.- 4. Experiment and Analysis.- 5. Results and Conclusions.- Appendix: Calculation of $$$$ in the SmA Phase.- Critical Fluctuations Under Shear Flow.- 1. Turbidity: TC change.- 2. The scattered light: Anisotropy mean-field lowering of UCD?.- 3. Discussion.- 4. "Moralite".- Lifshitz Points in Ising Systems with Competing Interactions.- 1. Introduction.- 2. One-dimensional Ising Systems with competing Interactions.- 3. Two-dimensional Ising systems.- 4. Conclusions.- Elementary Excitations in Magnetic Chains.- 1. Introduction.- 2. Magnons in XY-like Magnetic Chains.- 3. The Anisotropic, Classical XY Chain.- 3.1 The Model.- 3.2 Intuitive Analysis at Low Temperature for q = 2 and J = 0.- 3.3 Energy of a Wall.- 3.4 Number of Walls.- 3.5 Antiferromagnets in a Magnetic Field.- 4. A Simple Dynamical Model: The Almost - Ising Antiferromagnetic Chain.- 4.1 The Model.- 4.2 Collisions Between Two Solitons.- 5. Propagation of Broad Walls.- Experimental Studies of Linear and Nonlinear Modes in 1-D-Magnets.- 1. Real Systems Experimental Methods.- 2. Linear Excitations.- 3. Nonlinear Excitations.- Q-Dependence of the Soliton Response in CsNiF3 At.- T = 10K and H =5kG.- Dynamics of the Sine-Gordon Chain: The Kink-Phonon Interaction, Soliton Diffusion and Dynamical Correlations.- 1. Statement of the Problem.- 2. A Kink-Phonon Collision.- 3. Diffusive Motion of the Kink.- 4. Dynamical Correlation Functions.- The Spin-Wave Continuum of the S=1/2 Linear Heisenberg Antiferrornagnet.- Excitations and Phase Transitions in Random Anti-Ferromagnets.- Neutron Scattering.- Critical Phenomena at Phase Transitions.- Percolation.- Excitations of Dilute Magnets Near the Percolation Threshold.- Critical Properties of the Mixed Ising Ferromagnet.- Structure and Phase Transitions in Physisorbed Monolayers.- History and Background.- Statistical Thermodynamics of Physical Adsorption.- Structural Investigations of Monolayers.- Substrate Influences.- Commensurate-Incommensurate Transition and Orientational Epitaxy.- Antiferromagnetism in 0" Films.- Conclusion.- Two-Dimensional Solids and Their Interaction with Substrates.- I. Collective Phenomena and Phase Transitions in Two Dimensions.- 1.1 Early Theoretical Works.- 1.2 Experimental Situation.- II. Effect of Substrate.- II.1 Two-Dimensional Solids and Adsorbed Layers.- II.2 Substrate Distortion and Related Effects.- II.3 Chemical Potential.- II.4 Substrate Potential.- II.5 Conclusion.- III. Walls and Domains.- III.1 A One-Dimensional Model.- III.2 The Theory of Frank and Van der Merwe.- III.3 Aubry's Theory.- III.4 Domains and Walls for Dimensions Larger than 1.- IV. The Pokrovskii-Talapov Model.- IV.1 Hypotheses.- IV.2 Solution.- IV.3 Bragg Singularities.- IV.4 The Pinning Transition.- V. Rate Gas Monolayers on Graphite or Lamellar Halides.- V.1 Introduction.- V.2 The Zero Temperature Theory of Bak, Mukamel, Villain and Wentowska.- V.3 Rare Gas Monolayers on Hexagonal Substrates at T i 0.- V.4 Effect of Substrate Distortions.- VI. The Novaco-Mc Tague Orientational Instability.- VI.1 General Argument.- VI.2 Case of Parallel Walls.- VI.3 Case of a Regular Network of Intersecting Walls.- VI.4 Finite Temperatures.- VI.5 Microscopic Theories.- Appendix A. Bragg Singularities of a 2-D, Harmonic Crystal.- Appendix B. Interaction between two Solutions.- Appendix C. Partition function of the Pokrovskii-Talapov Model Near the Commensurable-Incommensurable Transition.- Appendix D. Bragg Singularities of the Pokrovskii-Talapov Model Near the C-I Transition.- Appendix E. The Roughening Transition.- The Dislocation Theory of Melting: History, Status and Prognosis.- 1. Introduction.- 2. History.- 3. Status.- 4. Prognosis.- The Kosterlitz-Thouless Theory of Two-Dimensional Melting.- Phase Transitions and Orientational Order in a Two-Dimensional Lennard-Jones System.- The Roughening Transition.- I. Introduction.- II. The solid-On-Solid Model.- III. The BCF Argument.- IV. Experimental Results.- V. Monte Carlo Calculations: Qualitative Features.- VI. Static Critical Behavior.- VII. Roughening Dynamics and the Kosterlitz Renormalization Group Method.- VIII.The FSOS Model and Mc Calculations.- IX. Final Remarks.- Statics and Dynamics of the Roughening Transition: A Self-Consistent Calculation.- I. Introduction.- II. Roughening Transition.- III. Two-Dimensional Planar Model.- IV. Conclusions.- Fluctuations in Two-Dimensional Six-Vertex Systems.- Light Scattering Studies of the Two-Dimensional Phase Transition in Squaric Acid.- 1. Introduction.- 2. Light Scattering Studies.- 3. Order Parameter.- 4. Order Flucatuations.- 5. Peak Shape and Width.- 6. Disorder Induced Scattering.- 7. Conclusions.- Monte Carlo Simulation of Dilute Systems and of Two-Dimensional Systems.- I. Introduction.- II. Ferromagnets Diluted with Nonmagnetic Impurities and Related Systems.- III. Models for Quasi-Two-Dimensional (2D) Magnets.- IV. Lattice Gas Models for Adsorbed Monolayers at Surfaces.- Order and Fluctuations in Smectic Liquid Crystals.- I. Introduction.- II. The Nematic Phase.- III.The Nematic-Smectic A Transition and The Smectic A Phase.- IV. The Smectic C Phase and Smc-SmA Transition.- v. Liquid Crystals and Lower Dimensional Physics.- Dislocations and Disclinations in Smectic Systems.- Translational Defects.- Orientational Defects.- Non-Elementary Defects.- Observation of Dislocations.- Dislocation Motion.- "Pair Creation" of Disclinations.- Defects and Phase Transitions.- Fluctuations and Freezing in a One-Dimensional Liquid:Hg3-?AsF6.- The Model Hamiltonian.- High Temperature Properties (T > TC).- Long Range Order.- Dynamics.- The Effect of Pressure on the Modulated Phases of TTF-TCNQ.- Spin Glasses A Brief Review of Experiments, Theories, and Computer Simulations.- 1. Spin Glass Materials and Experiments.- 2. Theoretical Models and Concepts.- 3. Spin-Glass Freezing: Phase Transition or Nonequilibrium Effect?.- 4. Conclusions and Outlook.- Random Anisotropy Spin-Glass.- Exact Results for a One-Dimensional Random-Anisotropy Spin Glass.- On Critical Slowing-down in Spin Glasses.- Participants.

Phase diagrams and critical behavior in Ising square lattices with nearest- and next-nearest-neighbor interactions

Abstract: The phase diagrams of Ising antiferromagnets in a magnetic field $H$ are investigated for various values of the ratio $R$ between nearest- and next-nearest-neighbor interaction. While meanfield approximations and the existing real-space renormalization-group treatments yield phase diagrams which are sometimes even qualitatively incorrect, accurate results are obtained from Monte Carlo calculations. For $Rl0$ only an antiferromagnetically ordered phase exists. Its transition to the disordered phase is first order for temperatures below the tricritical point (${T}_{t}$,${H}_{t}$). For $R\ensuremath{\rightarrow}0$ also ${T}_{t}\ensuremath{\rightarrow}0$. For $R=0$ we find very good agreement with the results of M\"uller-Hartmann and Zittartz. For $Rg0$ and ${H}_{1}lHl{H}_{2}$ a new phase with anomalous high ground-state degeneracy is found (two sublattices have only one-dimensional order). These sublattices undergo order-disorder transitions at $T=0$, such that for $Tg0$ one is left with a "superantiferromagnetic" phase. At low temperatures in this phase a pronounced tendency is observed to form a simpler (2 \ifmmode\times\else\texttimes\fi{} 2) superstructure but with many antiphase domain boundaries. For $R\ensuremath{\rightarrow}\frac{1}{2}$ and $Hl{H}_{1}$ the regime of the antiferromagnetic phases goes to zero temperature, while for $Rg\frac{1}{2}$ the superantiferromagnetic phase exists also for $Hl{H}_{1}$. The order-disorder transition associated with this phase seems to have non-Ising critical exponents which vary as a function of $R$ and $H$. Estimates for the exponents lead us to suggest that Suzuki's "weak universality" is valid. The behavior of the model at $T=0$ is related to known results on hard-core lattice gases. It is shown that it is useful to interpret the transitions at $T=0$ as generalized percolation transitions. Since the model may have applications to adsorbate phases in registered structures at (100) surfaces of cubic crystals, the transcription of our results to temperature-coverage phase diagrams and adsorption isotherms is discussed in detail, and possible experimental applications are mentioned.

Ordering in Two Dimensions

Abstract: During the last few years, the study of phase transitions in two dimensional systems has absorbed a great deal of effort by both theorists and experimentalists. Although such an activity is rather esoteric in the sense that these systems are rather special and do not occur in every day life, they are a theorist’s paradise because they form a very special class of systems for which theory is capable of yielding quantitative predictions. With the increase of the sensitivity of experiments, these predictions can now be checked by the experimentalists.

Normal fluctuations and the FKG inequalities

Abstract: In a translation invariant pure phase of a ferromagnet, finite susceptibility and the FKG inequalities together imply convergence of the block spin scaling limit to the infinite temperature Gaussian fixed point. This result is presented in a rather general probabilistic context and is applicable to infinite cluster density fluctuations in percolation models and to boson field fluctuations in (Euclidean) Yukawa quantum field theory models as well as to magnetization fluctuations in Ising models.

Solvable Model with a Roughening Transition for a Planar Ising Ferromagnet

Abstract: An exactly solvable modification of the planar Ising ferromagnet is proposed which has a roughening transition below the Curie temperature. The computation confirms the de Gennes-Fisher scaling theory of correlations with homogeneous surface fields, giving an exponent value ${\ensuremath{\Delta}}_{1}=\frac{1}{2}$.

Hard-square lattice gas

Abstract: We have studied the hard-square lattice gas, using corner transfer matrices. In particular, we have obtained the first 24 terms of the high-density series for the order parameterρ 2−ρ 1. From these we estimate the critical activity to be 3.7962±0.0001. This is in excellent agreement with the earlier work of Gaunt and Fisher. It conflicts with the value 4.0 given by Muller-Hartmann and Zittartz's formula for the critical point of the antiferromagnetic Ising model in a field, so we conclude that this formula, while a good approximation, is not exact.

Non-linear dynamics of classical one-dimensional antiferromagnets

Abstract: Solutions are presented to the non-linear equations of motion of classical antiferromagnetic chains in a continuum description. Results have been obtained for, in addition to isotropic exchange, various combinations of single-ion anisotropies (Ising and xy-like) and external magnetic fields (supporting and breaking the anisotropy). Among the solutions are both sine-Gordon solitons, representing antiferromagnetic domain walls and pulse solitons with continuously varying amplitude. Solitons in the xy antiferromagnet in asymmetry-breaking external field are discussed with respect to their observability in TMMC. They are found to contribute two different central peaks to the dynamical structure factor.

Ordering of the Face-Centered-Cubic Lattice with Nearest-Neighbor Interaction

Abstract: The phase diagram of an Ising fcc antiferromagnet in a field is investigated by a study of the ground state and Monte Carlo simulation of a lattice with 16 384 sites. At the critical field between the two "ordered" phases, the disordered phase is stable down to zero temperature due to "frustration" effects. The corresponding alloy phase diagram disagrees with all previous calculations. Order parameters and internal energy are briefly compared to experimental data on the Cu-Au system.

Analytic theory of the ground state properties of a spin glass. I. Ising spin glass

Abstract: For pt.I see ibid., vol.10, no.12, p.2769 (1980). The general theory, developed in the preceding paper to study the ground state properties of spin glasses, is applied to two-component spin systems. As an extension of the Edwards-Anderson model (1975) a system of the two-component spins, each spin of which can take only n directions in a plane ('discrete planar model'), is introduced to count (gn)-the average degeneracy in the ground state. This procedure allows the authors to see how the fundamental properties change by the spin symmetry. For n=2, the theory reproduces the previous results for the Ising model. They can study the XY model as a limit of n to infinity . Precise calculations are performed for the infinite-range exchange interactions. They find (ginfinity )=e0.51691N.

The use of anticommuting variable integrals in statistical mechanics. II. The computation of correlation functions

Abstract: Integrals over anticommuting variables are use to rewrite partition functions as fermionic field theories. The method is used to solve the two‐dimensional Ising model, the planar close‐packed dimer problems, and the free‐fermion eight vertex model.

White and weighted averages over solutions of Thouless Anderson Palmer equations for the Sherrington Kirkpatrick spin glass

Abstract: 2014 We study white and weighted averages of observables over all solutions of Thouless Anderson and Palmer equations. For the white average we give analytic results near the transition temperature and near T = 0, in good qualitative agreement with expected results for the Edwards Anderson order parameter. A too high value for the ground state energy is attributed to the non discriminating character of the white average. Local stability in all directions is verified for the relevant saddle point (diagonal in replica space). We also study a microcanonical average at T = 0. In that case, the diagonal saddle point is unstable and equations for the off-diagonal one are not soluble by purely analytical means. Tome 41 N° 9 SEPTEMBRE 1980 J. Physique 41 (1980) 923-930 SEPTEMBRE 1980, Classification Physics Abstracts 75.10H

Two-dimensional ising models with competing interaction—a Monte Carlo study

Abstract: Two-dimensional Ising models on a square lattice with competing interactions along one axis or both axes are studied primarily by the Monte Carlo method. Several commensurate-incommensurate transitions are found. Dislocation-like configurations are identified near the sinusoidal — paramagnetic transition in accordance with the idea that the transition might be of Kosterlitz-Thouless, XY-like character.

Exact results and critical properties of the Ising model with competing interactions

Abstract: The random Ising model with competing interactions is investigated on the basis of the gauge-invariant formulation of the problem. Exact results for the internal energy, specific heat and gauge-invariant correlation function are derived. The critical exponent alpha is shown to be negative at the phase boundary of the paramagnetic and ferromagnetic phases if the latter exists at fairly low concentration of antiferromagnetic bonds.

Wall effects in critical systems: Scaling in Ising model strips

Abstract: A scaling theory and a related conjecture presented recently by de Gennes and Fisher to predict the effects of walls inserted in two near-critical binary fluid mixtures, are checked theoretically by exact analytic calculations for $n\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$, two-dimensional Ising model strips with a surface field, ${h}_{1}$, imposed on the first layer; exact calculations with a second surface field, ${h}_{n}$, imposed on the nth layer are also reported. It is verified, in particular, that the effects on properties observed close to one wall of a second wall at distance $D$ decay, at the critical point, as $\frac{1}{{D}^{d}}$, where $d$ is the spatial dimensionality. In addition to the scaling limits, the leading corrections are calculated explicitly and presented graphically.

Nonlinear Partial Difference Equations for the Two-dimensional Ising Model

Abstract: The two-point function of the two-dimensional Ising model at arbitrary temperature is expressed in terms of the solution of a nonlinear partial difference equation. From this difference equation the known results for the two-point function of the Ising field theory may be regained as a special case.

Finite-lattice approach to the O(2) and O(3) models in 1 + 1 dimensions and the (2+1)-dimensional Ising model

Abstract: The Lanczos scheme for finding low-lying eigenvalues of a sparse matrix of large dimension is applied to solving the Hamiltonian formulation for the O(2) and O(3) spin systems in 1+1 dimensions and the Ising model in 2+1 dimensions. We confirm results obtained for these models in other ways. The new method is shown to be competitive with the other methods available for solving these problems.

Phase boundary of Ising antiferromagnets near H = H c and T = 0 : Results from hard-core lattice gas calculations

Abstract: Ising antiferromagnets in a near-critical magnetic field at low temperatures are equivalent to hard-core lattice gases. Using this connection and the existing series-expansion results for hard-core lattice gases, we determine the slope of the phase boundary at $T=0$ for the square (sq), plane-triangular (pt), simple cubic (sc), and body-centered cubic (bcc) antiferromagnets. The slope is negative for the sq and pt lattices and nearly zero for the sc case. For the bcc lattice a positive slope is obtained, indicating that the phase boundary bulges above the zero-temperature critical field. We also test M\"uller-Hartmann and Zittarz's postulate for the critical curve of the sq Ising antiferromagnet. A renormalization-group treatment of the hard-square lattice gas yields a critical activity ${z}^{*}=3.7959\ifmmode\pm\else\textpm\fi{}0.0001$, which is in agreement with series-expansion and finite-lattice estimates but at variance with the postulated ${z}^{*}=4$. The same calculation gives $\ensuremath{ u}=0.999\ifmmode\pm\else\textpm\fi{}0.001$ for the correlation-length exponent, thus supporting the conjecture that the transition of the hard-square lattice gas belongs to the Ising universality class.

Renormalisation group calculations on a mixed-spin system in two dimensions

Abstract: The real space renormalisation group of Niemeijer and van Leeuwen (1974) is applied to a mixed-spin Ising model on a simple quadratic lattice. The motivation is the investigation of critical phenomena in Ising models with less than the usual translational symmetry. The models in question are relevant to the study of ferrimagnetism. Two calculations, characterised by different block constructions, are performed and compared. Exponent values are found to be in good agreement with those suggested by the universality hypothesis. The utility of the renormalisation group for dealing with ferrimagnetism is demonstrated, but the high degree of labour involved in such an exercise is indicated.