Showing papers on "Ising model published in 1982"
TL;DR: In this article, it was shown that the φ4 Euclidean field theory with a lattice cut-off is inevitably free in the single phase regime in ind>4 dimensions, and that the critical behavior in Ising models is in exact agreement with the mean-field approximation in high dimensions, but not in the low dimensiond=2.
Abstract: We provide here the details of the proof, announced in [1], that ind>4 dimensions the (even) φ4 Euclidean field theory, with a lattice cut-off, is inevitably free in the continuum limit (in the single phase regime). The analysis is nonperturbative, and is based on a representation of the field variables (or spins in Ising systems) as source/sink creation operators in a system of random currents — which may be viewed as the mediators of correlations. In this dual representation, the onset of long-range-order is attributed to percolation in an ensemble of sourceless currents, and the physical interaction in the φ4 field — and other aspects of the critical behavior in Ising models — are directly related to the intersection properties of long current clusters. An insight into the criticality of the dimensiond=4 is derived from an analogy (foreseen by K. Symanzik) with the intersection properties of paths of Brownian motion. Other results include the proof that in certain respect, the critical behavior in Ising models is in exact agreement with the mean-field approximation in high dimensionsd>4, but not in the low dimensiond=2 — for which we establish the “universality” of hyperscaling.
393 citations
TL;DR: In this article, it was shown rigorously that the one-dimensional Ising model with long-range antiferromagnetic interactions exhibits a complete devil's staircase, which is the same as the one in this paper.
Abstract: It is shown rigorously that the one-dimensional Ising model with long-range antiferromagnetic interactions exhibits a complete devil's staircase.
330 citations
TL;DR: The partition-functions-per-site k-approximation of several two-dimensional models (notably the eight-vertex, self-dual Potts and hard-hexagon models) can be easily obtained by using an inversion relation for local transfer matrices, together with symmetry and analyticity properties as discussed by the authors.
Abstract: The partition-functions-per-siteκ of several two-dimensional models (notably the eight-vertex, self-dual Potts and hard-hexagon models) can be easily obtained by using an inversion relation for local transfer matrices, together with symmetry and analyticity properties. This technique is discussed, the analyticity properties compared, and some equivalences (and nonequivalences) pointed out. In particular, the critical hard-hexagon model is found to have the sameκ as the self-dualq-state Potts model, withq=(3 + √5)/2 = 2.618 .... The Temperley-Lieb equivalence between the Potts and six-vertex models is found to fail in certain nonphysical antiferromagnetic cases.
277 citations
TL;DR: In this paper, a numerical method was developed to study the scaling of distribution of couplings of highly random antiferromagnetic Ising and quantum Heisenberg spin-\textonehalf{} systems.
Abstract: A numerical method is developed to study the scaling of distribution of couplings of highly random antiferromagnetic Ising and quantum Heisenberg spin-\textonehalf{} systems. The method shows how freezing into inert local singlets prevents ordering down to temperatures well below the median nearest-neighbor coupling or bare exchange percolation threshold in positionally disordered systems with Heisenberg exchange varying exponentially with distance (e.g., doped semiconductors, quasi one-dimensional salts). This is contrasted with the Ising system.
275 citations
01 Jan 1982
TL;DR: In this paper, a review of real-space Renormalization methods is presented, which includes the following: 1.1.1 Bond-Moving and Variational Methods.- 1.2.2 Inclusion of a Magnetic Field.- 2.3.4 Variational Transformations.
Abstract: 1. Progress and Problems in Real-Space Renormalization.- 1.1 Introduction.- 1.2 Review of Real-Space Renormalization.- 1.3 New Renormalization Methods.- 1.3.1 Bond-Moving and Variational Methods.- 1.3.2 Monte Carlo Renormalization.- 1.3.3 Exact Differential Transformations.- 1.3.4 Phenomenological Renormalization.- 1.4 New Applications.- 1.4.1 Adsorbed Systems.- 1.4.2 Applications to Quantum Systems.- 1.4.3 Percolation and Polymers.- 1.4.4 Dynamic Real-Space Renormalization.- 1.4.5 The Kosterlitz-Thouless Transition.- 1.4.6 Field-Theoretical Applications.- 1.5 Fundamental Problems.- 1.5.1 Choice of the Weight Function.- 1.5.2 Griffiths-Pearce Peculiarities.- 1.6 Exact Differential Real-Space Renormalization.- 1.6.1 The Two-Dimensional Ising Model.- 1.6.2 Discussion.- 1.7 Phenomenological Renormalization.- 1.7.1 Description of the Method.- 1.7.2 Applications.- 1.8 Concluding Remarks.- References.- 2. Bond-Moving and Variational Methods in Real-Space Renormalization.- 2.1 Introduction.- 2.2 Variational Principles.- 2.2.1 Lower-Bound Property of Bond-Moving Approximations.- 2.2.2 Upper-Bound Property of the First-Order Cumulant Approximation.- 2.3 The Migdal-Kadanoff Transformation.- 2.3.1 Application to the Ising Model with Nearest-Neighbor Interactions.- 2.3.2 Inclusion of a Magnetic Field.- 2.3.3 The Bond-Moving Prescription of EMERY and SWENDSEN.- 2.3.4 Inconsistent Scaling of the Correlation Function.- 2.3.5 Relation to Exactly Soluble Hierarchical Models.- 2.3.6 Applications.- 2.3.7 Modifications of the Migdal-Kadanoff Procedure.- 2.4 Variational Transformations.- 2.4.1 The Kadanoff'Lower-Bound Variational Transformation.- 2.4.2 The Kadanoff Criterion for the Optimal Variational Parameter.- 2.4.3 Problems with the Lower-Bound Variational Transformation.- 2.4.4 Determination of an Optimal Sequence of Variational Parameters.- 2.4.5 Applications of the Lower-Bound Variational Transformation.- 2.4.6 Other Variational Methods.- 2.5 Conclusion.- References.- 3. Monte Carlo Renormalization.- 3.1 Introduction.- 3.2 Basic Notation and Renormalization-Group Formalism.- 3.3 Large-Cell Monte Carlo Renormalization Group.- 3.4 MCRG.- 3.4.1 Calculation of Critical Exponents.- 3.4.2 Calculation of Renormalized Coupling Constants.- 3.5 MCRG Calculations for Specific Systems.- 3.6 Other Approaches to the Monte Carlo Renormalization Group.- 3.7 Conclusions.- References.- 4. The Real-Space Dynamic Renormalization Group.- 4.1 Introduction.- 4.2 Dynamic Problem of Interest.- 4.3 RSDRG - Formal Development.- 4.4 Implementation of the RSDRG Using Perturbation Theory.- 4.4.1 General Development.- 4.4.2 Expansion for H and D?.- 4.4.3 Solution to the Zeroth-Order Problem.- 4.4.4 Renormalization to First Order.- 4.4.5 Recursion Relations for the Correlation Functions.- 4.5 Determination of Parameters.- 4.5.1 General Comments.- 4.5.2 The Parameters K0 and KR0.- 4.5.3 The Dynamic Parameters ?0 and ?.- 4.6 Results.- 4.7 Discussion.- References.- 5. Renormalization for Quantum Systems.- 5.1 Background.- 5.2 Application of the Niemeijer-van Leeuwen Renormalization Group Method to Quantum Lattice Models.- 5.3 The Block Method.- 5.3.1 Principles.- 5.3.2 Applications.- a) The Ising Model in a Transverse Field in One Dimension.- b) The Free Fermion Model in One Dimension.- 5.3.3 Extensions of the Method.- a) Extension to Large Blocks.- b) Extension by Increasing the Number nL of Levels Retained.- c) Other Extensions.- 5.4 Applications of the Block Method.- 5.4.1 Spin Systems.- a) The Spin 1/2 Ising Model in a Transverse Field (ITF).- b) The XY Heisenberg Spin 1/2 Chain.- c) The XY Model in a Z Field for d = 2, 3.- d) The Spin 1 XY Model with an Anisotropy Field for d = 1.- 5.4.2 Fermion Systems.- a) The d = 1 Hubbard Model.- b) Interacting Fermions in d = 1.- c) One-Dimensional Model of f and d Electrons with Hybridization V and fd Interaction Ufd.- 5.4.3 Spin Fermion Systems: The Kondo Lattice in d = 1.- 5.4.4 Quantum Versions of Classical Statistical Mechanics in 1 + 1 Dimension.- a) The 0(n) Model.- b) The P(q) Potts Model.- c) Tricritical Point for Ising Systems in 1 + 1 Dimensions.- 5.4.5 Applications to Field Theory.- a) The Thirring Model in One-Space and One-Time Dimension.- b) The U(1) Goldstone Model in Two Dimensions.- c) Lattice Gauge Theories.- 5.5 Discussion.- 5.5.1 When is the BRG More Suitable?.- 5.5.2 How to Control the Method?.- a) The Division of the Lattice into Blocks.- b) Which Levels to Retain for the Truncated Basis?.- 5.5.3 What Has Been Done and What Are the Difficulties Encountered?.- a) Quantum Properties at T = 0.- b) Quantum Properties at T ? 0.- c) Difficulties.- 5.5.4 Comparison Between Different Methods.- a) The Real-Space RG Methods for Classical Systems.- b) Finite-Size Scaling Methods.- 5.6 What to Do Next?.- 5.6.1 Improvement of the Method.- 5.6.2 Applications.- References.- 6. Application of the Real-Space Renormalization to Adsorbed Systems.- 6.1 Introduction.- 6.2 The Sublattice Method.- 6.3 The Prefacing Method and Introduction of Vacancies.- 6.4 The Potts Model.- 6.5 Further Applications of the Vacancy.- 6.6 Summary.- References.- 7. Position-Space Renormalization Group for Models of Linear Polymers, Branched Polymers, and Gels.- 7.1 Three Physical Systems.- 7.1.1 Linear Polymers.- 7.1.2 Branched Polymers.- 7.1.3 Gels.- 7.2 Three Mathematical Models.- 7.2.1 Percolation.- 7.2.2 Self-Avoiding Walks.- 7.2.3 Lattice Animals.- 7.3 Position-Space Renormalization Group Treatment.- 7.3.1 Percolation.- a) Basic Approach.- b) Extensions.- 7.3.2 Self-Avoiding Walks.- a) Basic Approach.- b) Extensions.- 7.3.3 Lattice Animals.- a) Basic Approach.- b) Extensions.- 7.4 Other Approaches.- 7.4.1 Percolation.- 7.4.2 Self-Avoiding Walks.- 7.4.3 Lattice Animals.- 7.5 Concluding Remarks and Outlook.- References.
248 citations
TL;DR: In this article, the effects of dislocations on the two-dimensional order of the modulated phases of a uniaxial ferromagnet were investigated using a Ginzburg-Landau approach.
Abstract: A uniaxial ferromagnet of finite thickness spontaneously breaks into parallel "striped" domains as a result of demagnetizing forces. In a sufficiently large applied magnetic field a transition takes place to a hexagonal or "bubble" phase. We study the phase diagram of this system with the use of a Ginzburg-Landau approach and we investigate the effects of dislocations on the two-dimensional order of the modulated phases.
246 citations
TL;DR: In this article, the existence of spontaneous magnetization at low temperature for the one-dimensional Ising model with 1/r2 interaction energy was proved, and it was shown that the magnetization can be obtained at low temperatures.
Abstract: We prove the existence of a spontaneous magnetization at low temperature for the one-dimensional Ising Model with 1/r2 interaction energy.
245 citations
192 citations
TL;DR: In this paper, the authors review finite-size scaling theory from the general point of view of renormalization group theory for both continuous and first-order transitions (both for systems with discrete and continuous symmetries).
Abstract: Research in recent years has shown that combining finite‐size scaling theory with the transfer matrix technique yields a powerful tool for the investigation of critical behavior. In particular, the method has been used to study two‐dimensional statistical mechanical and one‐dimensional quantum mechanical systems. We review finite‐size scaling theory from the general point of view of renormalization group theory for both continuous and first‐order transitions (both for systems with discrete and continuous symmetries). We review applications where a comparison with exact results can be made. These include the Ising, Baxter, and q‐state Potts models and the Ising model with a defect line. Various other applications such as quantum systems, the self‐avoiding random walk, percolation, and Kosterlitz‐Thouless transitions are briefly reviewed. The Kosterlitz‐Thouless transitions and the critical fan in the antiferromagnetic 3‐state Potts model are discussed at somewhat greater length.
167 citations
TL;DR: In this article, the zero-temperature triangular Ising antiferromagnet is mapped onto a solid-on-solid (SOS) model, and the system undergoes a roughening transition characterised by a critical exponent alpha = 1/2, by the absence of excitations in the smooth phase, and by domain wall excitations (stripes) in the rough phase.
Abstract: The zero-temperature triangular Ising antiferromagnet is mapped onto a solid-on-solid (SOS) model. The system undergoes a roughening transition characterised by a critical exponent alpha =1/2, by the absence of excitations in the smooth phase, and by domain wall excitations (stripes) in the rough phase. At infinite SOS temperature the height-height correlation function is explicitly calculated with the aid of known four-point Ising correlations. The authors point out that a certain six-vertex model with a comparable SOS interpretation has an identical critical temperature, critical exponent and critical amplitude. This is in support of existing ideas on university in systems with striped phases.
157 citations
TL;DR: In this paper, the lower critical dimension of random-field Ising systems is shown to be two, in agreement with simple domain estimates, based on a new interface model and a simple domain model.
Abstract: It is argued on the basis of a new interface model that the lower critical dimension of random-field Ising systems is two, in agreement with simple domain estimates.
TL;DR: Mean field theory for one-dimensional inhomogeneous magnetic systems is formulated as an area-preserving map and its associated boundary conditions are derived for nearest-neighbor Ising interactions as discussed by the authors.
Abstract: Mean-field theory for one-dimensionally inhomogeneous magnetic systems is formulated as an area-preserving map. The map and its associated boundary conditions are derived for nearest-neighbor Ising interactions. The corresponding continuum theory is also constructed. These mappings are two dimensional. Their phase portraits are exhibited and applied to the study of a representative set of surface and interface phenomena, including interfacial structure, surface phase transitions, wetting, prewetting, and layering. The methods developed lend themselves to easy and physical visualization of the types of solutions which the mean-field theory can have, even in rather complex situations. They also make explicit the fundamental differences between continuum mean-field theory (which is integrable) and discrete mean-field theory (which is not).
TL;DR: In this article, the scaling-field method was used to calculate critical properties of continuous-spin Landau Hamiltonians of cubic symmetry, such as the spin dimensionality and the number of spin components.
Abstract: Critical properties of models defined by continuous-spin Landau Hamiltonians of cubic symmetry are calculated as functions of spatial dimensionality, $2.8\ensuremath{\le}d\ensuremath{\le}4$, and number of spin components, $N$. The investigation employs the scaling-field method developed by Golner and Riedel for Wilson's exact momentum-space renormalization-group equation. Fixed points studied include the isotropic and decoupled Ising ($\ensuremath{-}2\ensuremath{\le}Nl\ensuremath{\infty}$), the face-and corner-ordered cubic ($1\ensuremath{\le}Nl\ensuremath{\infty}$), and, via the replica method for $N\ensuremath{\rightarrow}0$, the quenched random Ising fixed point. Variations of $N$ and $d$ are used to link the results to exact results or results from other calculational methods, such as $\ensuremath{\epsilon}$ expansions near two and four dimensions. This establishes the consistency of the calculation for three dimensions. Specifically, truncated sets involving seven (twelve) scaling-field equations are derived for the cubic $N$-vector model. A stable random Ising fixed point is found and shown to be distinct from the cubic fixed point and to connect, as a function of $d$, with the Khmelnitskii ${\ensuremath{\epsilon}}^{\frac{1}{2}}$ fixed point. At $d=3$, the short truncation yields $\ensuremath{\alpha}\ensuremath{\approx}0.11$ for the pure Ising and $\ensuremath{\alpha}\ensuremath{\approx}\ensuremath{-}0.09$ for the random Ising fixed point. A search for a random tricritical fixed point was inconclusive. For the $N$-component cubic model, the spin dimensionality ${N}_{c}$, at which the isotropic and cubic fixed points change stability, is determined as a function of $d$. The results support ${N}_{c}g3$ for three dimensions.
TL;DR: In this article, the surface tension, the equilibrium shape of a crystal, and the total surface energy of a given crystal were analyzed for the $d=2$ Ising model with anisotropic couplings, and a discussion of the solid-on-solid approximation was presented.
Abstract: Explicit relations between the surface tension (interfacial energy density), the equilibrium shape of a crystal, and the total surface energy are given. For the $d=2$ Ising model with anisotropic couplings, the exact equilibrium shape is cast into a closed form of elementary functions. The surface energy is compared with Monte Carlo simulations. A discussion of the solid-on-solid approximation is presented.
TL;DR: In this paper, the exact partition function of the Ising model on a periodic 4-simple cubic lattice is presented, and the methods used in deriving this quantity, based on low-temperature series, are discussed.
Abstract: The exact partition function of the Ising model on a periodic 4\ifmmode\times\else\texttimes\fi{}4\ifmmode\times\else\texttimes\fi{}4 simple cubic lattice is presented. The methods used in deriving this quantity, based on low-temperature series, are discussed. The zero-field partition function is analyzed to locate its zeros in the complex plane, and these results are discussed.
TL;DR: In this article, an euclidean functional integral approach for the construction of local kink operators and disorder variables is proposed and investigated, where the main difference from the existing quasiclassical approach is the emphasis on local fields instead of kink states and collective coordinate methods.
TL;DR: In this article, a simple recurrence relation for successive shell magnetizations far removed from the surface has been established for the Ising model on a Cayley tree with zero long-range order at all finite temperatures.
Abstract: The Ising model on a Cayley tree displays a peculiar (continuous order) phase transition with zero long-range order at all finite temperatures. When one studies expection values of spins far removed from the surface (which contains a finite fraction of the total number of spins in the thermodynamic limit), however, one obtains the so-called Bethe approximation. Here we study such a local description by setting up a simple recurrence relation for successive shell magnetizations far removed from the surface. In the ferromagnetic case the local magnetization is a fixed point of the iterative transformation, while in the antiferromagnetic case the fixed point bifurcates to a two-cycle of the transformation (for low temperatures and fields) giving rise to local sublattice magnetizations. In both cases, local thermodynamical properties are obtained by integration.
TL;DR: In this paper, the results for the ferromagnetic system were compared with theoretical predictions derived from a bond-diluted Ising model using the replica method and a second-order variational approximation.
Abstract: Spin glass behaviour has been observed in a very large concentration range in both CdCr2xIn2-2xS4 and ZnCr2xAl2-24 spinel systems. It is a result of the presence in these systems of competing interactions, ferromagnetic between first-nearest neighbours and antiferromagnetic between higher-order neighbours. The ratios between the interactions are such that the non-dilute x=1 compounds are ferromagnetic for the first system and helimagnetic for the second. The entire phase diagrams of both solutions have been determined by very low field AC and DC susceptibility measurements. The results for the ferromagnetic system are compared with theoretical predictions derived from a bond-diluted Ising model using the replica method and a second-order variational approximation.
TL;DR: Dilute, random fields and random bonds ising systems on a d -dimensional hypercubic lattice are analyzed with a new approximate scheme for the study of disordered systems.
TL;DR: In this article, a simple derivation of a grassmannian integral for the 3D Ising model partition function is presented and some elementary consequences of the derivation are studied.
TL;DR: In this paper, inequalities relating to the interface between coexisting phases of Ising ferromagnets are discussed, and some implications for the nature of the roughening transition are discussed.
Abstract: We describe inequalities relating to the interface between coexisting phases of Ising ferromagnets. Some implications for the nature of the roughening transition are discussed.
TL;DR: In this article, a common construction of the three-dimensional equivalents of these symmetry relations is presented, which is used to derive, in a geometric way applying simultaneously to different kinds of spin models, the consequent global properties, i.e., the commutativity of the transfer matrices and the inverse functional equations on the transfer matrix and the partition function.
Abstract: Symmetry relations such as the star-triangle or the inverse relation are very useful in determining the partition function of two-dimensional exactly soluble models. A common construction of the three-dimensional equivalents of these symmetry relations is presented. They are used to derive, in a geometric way applying simultaneously to different kinds of spin models, the consequent global properties, i.e. the commutativity of the transfer matrices and the inverse functional equations on the transfer matrix and the partition function. The usefulness of the inverse relation is illustrated by an application to the three-dimensional Ising model.
TL;DR: In this paper, the authors discussed linear and nonlinear susceptibilities of spin glasses in thermal equilibrium and qualitatively discussed the apparent discrepancy between theoretical results, implying lack of Edwards-Anderson order and hence a divergence of the susceptibility at zero temperature, and experiments showing a susceptibility saturating at finite values.
Abstract: Linear and nonlinear susceptibilities of spin glasses in thermal equilibrium are qualitatively discussed in terms of magnetic short-range order. The apparent discrepancy between theoretical results, implying lack of Edwards-Anderson order and hence a divergence of the susceptibility at zero temperature, and experiments showing a susceptibility saturating at finite values, is considered. It is suggested that the susceptibility must indeed have a static maximum if the system exhibits a (“frustrated”) ferromagnetic phase with a reentrant phase boundary. At the reentrancy point, some exponents of the ferromagnet take twice their normal value, and hence a crossover near this point occurs similar to multicritical points. These observations are used to interpret a number of recent experiments, and it is shown that neither of them proves the existence of a static spin glass phase. As a quantitative example of gradual onset of order without a phase transition in three-dimensional systems, numerical results for susceptibility and specific heat of the fully frustrated Ising fcc antiferromagnet at its critical field are given.
TL;DR: In this article, the authors analyzed how random fields are generated in a site-random uniaxial antiferromagnet when a uniform magnet field is applied Explicit expressions are given for the mean square random field.
Abstract: We analyze in detail how random fields are generated in a site‐random uniaxial antiferromagnet when a uniform magnet field is applied Explicit expressions are given for the mean‐square random field We also report on a systematic specific heat study of two three‐dimensional Ising systems, pure FeCl2 and site‐random Fe0682Mg0318Cl2 (MG318), in constant applied fields The contrasting behavior of the two provide a vivid demonstration of random‐field effects The unusual changes in the shape of the specific heat anomaly observed in MG318 are explained as crossover behavior and not a manifestation of new critical behavior The crossover exponent φ associated with the random fields is determined from the shape of the magnetic phase boundary We find φ=125±010, in excellent agreement with theoretical prediction
TL;DR: In this article, the fermionization of the two-dimensional Ising model and its applications are reviewed, and the first part of a series devoted to Ising like systems is presented.