Showing papers on "Ising model published in 1979"

Contribution to the new type of effective-field theory of the Ising model

Abstract: A new type of effective-field theory of the Ising model is presented. The differential-operator method is introduced into the exact spin correlation function identity obtained by Callen. The Curie temperatures are evaluated by using two different types of effective Hamiltonians. It is also shown how the Zernike and the Bethe-Peierls equations can be reproduced within one framework depending on the choice of effective fields in an effective Hamiltonian. The spin correlation function and the specific heat are presented.

Experiments with a Gauge-Invariant Ising System

Abstract: Using Monte Carlo techniques, we evaluate the path integral for the four-dimensional lattice gauge theory with a Z/sub 2/ gauge group. The system exhibits a first-order transition. This is contrary to the implications of the approximate Migdal recursion relations but consistant with mean-field-theory arguments. Our ''data'' agree well with a low-temperature expansion and the exact duality between the high- and low-temperature phases.

Phase structure of discrete Abelian spin and gauge systems

Abstract: It is shown that in a two-dimensional ${Z}_{p}$ spin system for $p$ not too small there exists a massless phase in the middle between the ordered and disordered Ising-type phases. A similar thing happens in a four-dimensional ${Z}_{p}$ gauge theory, where a massless QED-like phase appears between the screened and the confined phases. The existence of the middle phase is deduced logically from the existence of such a phase in the continuous O(2)-invariant models using self-duality and correlation inequalities. For the spin case the transition towards this phase is analyzed using a Kosterlitz type of renormalization group suggesting an essential singularity of the correlation length at both transition points. A Hamiltonian strong-coupling expansion up to ninth order is applied to the ${Z}_{p}$ spin system. The results of the Pad\'e analysis of this expansion are consistent with the phase structure described above. For $p\ensuremath{\le}4$ the analysis suggests two phases with a conventional singularity behavior at the transition. In the nontrivial case of $p=3$, critical exponents are calculated and found to give good agreement with experiment. For $p\ensuremath{\ge}5$ the analysis favors three phases with an essential singularity at the transition.

Construction of Green's functions from an exact S matrix

Abstract: The bootstrap program for determining Green's functions from an exact $S$ matrix is carried out for the simplest soliton field theory of a scalar field with $S$-matrix operator $S={(\ensuremath{-}1)}^{\frac{N(N\ensuremath{-}1)}{2}}$, where $N$ is the total number operator. Despite the formal simplicity of the $S$ matrix, the Green's functions derived have a rich structure. The results can be checked since this field theory is none other than that of the order variable of the Ising model in the scaling limit above the critical temperature.

Mathematical properties of position-space renormalization-group transformations

Abstract: Properties of “position-space” or “cell-type” renormalization-group transformations from an Ising model object system onto an Ising model image system, of the type introduced by Niemeijer, van Leeuwen, and Kadanoff, are studied in the thermodynamic limit of an infinite lattice. In the case of a KadanofF transformation with finitep, we prove that if the magnetic field in the object system is sufficiently large (i.e., the lattice-gas activity is sufficiently small), the transformation leads to a well-defined set of image interactions with finite norm, in the thermodynamic limit, and these interactions are analytic functions of the object interactions. Under the same conditions the image interactions decay exponentially rapidly with the geometrical size of the clusters with which they are associated if the object interactions are suitably short-ranged. We also present compelling evidence (not, however, a completely rigorous proof) that under other conditions both the finite- and infinite-p (“majority rule”) transformations exhibit peculiarities, suggesting either that the image interactions are undefined (i.e., the transformation does not possess a thermodynamic limit) or that they fail to be smooth functions of the object interactions. These peculiarities are associated (in terms of their mathematical origin) with phase transitions in the object system governed not by the object interactions themselves, but by a modified set of interactions.

Evidence Against Spin-Glass Order in the Two-Dimensional Random-Bond Ising Model

Abstract: By recursive methods, numerically exact free energies are calculated for L×L Ising lattices with bonds of randomly chosen sign, with 6<~L<~18. Ground states of these systems are identified, and the response to ordering fields is studied. By performing Monte Carlo simulations for precisely the same systems we are able to unambiguously distinguish nonequilibrium phenomena from equilibrium properties. The L dependence of our results suggests that there is no nonzero spin-glass order parameter for L→∞.

Zero-temperature renormalization method for quantum systems. I. Ising model in a transverse field in one dimension

Abstract: A zero-temperature real-space renormalization-group method is presented and applied to the quantum Ising model with a transverse field in one dimension. The transition between the low-field and high-field regimes is studied. Magnetization components, spin correlation functions, and critical exponents are derived and checked against the exact results. It is shown that increasing the size of the blocks in the iterative procedure yields more accurate results, especially for the critical ''magnetic'' exponents near the transition.

Monte Carlo Renormalization Group and Ising Models with n>=2

Abstract: We suggest that the Monte Carlo renormalization group, when combined with the type of cell-spin transformation introduced by van Leeuwen, should be a powerful tool in the study of Ising models with n > or = 2. Numerical results are presented for the Baxter model and the Ising model with nearest-- and next-nearest--neighbor interactions on a square lattice.

Hamiltonian approach to Z (N) lattice gauge theories

Abstract: We develop a Hamiltonian formalism for $Z(N)$ lattice gauge theories. Duality is expressed by algebraic operator relations which are the analog of the interchange of electric and magnetic fields in $D=3$ space dimensions. In $D=2$ duality is used to solve the gauge condition. This leads to a generalized Ising Hamiltonian. In $D=3$ our theory is self-dual. For $N\ensuremath{\rightarrow}\ensuremath{\infty}$ the theory turns into "periodic QED" in appropriate limits. This leads us to propose the existence of three phases for $Ng{N}_{c}\ensuremath{\simeq}6$. Their physical properties can be classified as electric-confining, nonconfining, and magnetic-confining.

Euclidean Group as a Dynamical Symmetry of Surface Fluctuations: The Planar Interface and Critical Behavior

Abstract: The statistical mechanics of the interface between two discrete thermodynamic phases is studied in terms of a field f which describes the deviation of the interface from planar. Exploiting the dynamical Euclidean invariance of the Hamiltonian of the field f, we construct epsilon expansions for Ising-like critical behavior in 1+epsilon dimensions, with a critical temperature of order epsilon. Scaling functions for the interface profile and width in a pinning potential are obtained.

Conjecture on the exact transition point of the random Ising ferromagnet

Abstract: Presents a conjecture on the exact value of the transition point of the random Ising ferromagnet on the square lattice. Only the bond problem is treated; each neighbouring pair of spins has an interaction of strength J(>or=0), where J is a random variable with an arbitrarily given probability distribution. To solve the problem an exact duality transformation and the replica method are used. In order to deduce the transition point from the duality relation only it is necessary to make an unconfirmed assumption concerning the symmetry of the distribution of singularities of the free energy.

Internal field distributions in model spin glasses

Abstract: The zero temperature probability distribution P(h) of internal magnetic fields is studied, both in the Sherrington-Kirkpatrick random Ising model of a spin glass and in its natural extension to classical vector spins. Theoretical predictions and computer simulations agree that P(h) is linear for small h in the Ising case, and has a hole-P(h)=0 for h< eta -in the vector spin case.

Frustration in periodic systems : exact results for some 2D Ising models

Abstract: We present exact results for various frustrated Ising models in two dimensions with periodic interactions. None of them gives rise to what could be regarded as spin glass order. The only long range orders encountered are of ferro- or antiferromagnetic type but the effect of frustration manifests itself at low temperature (entropy, susceptibilities). The analysis of these results provides some indications concerning a necessary condition for ordering in such systems and the possible nature of the spin glass order.

Dynamic properties of a spin-glass model at low temperatures

Abstract: We present a semiphenomenological calculation of the low-temperature dynamic properties of a spin-glass model which is a two-dimensional Ising model with Gaussian random nearest-neighbor interactions. The distribution of the low-lying energy levels of the system is studied with the aid of a numerical program. The results of this investigation suggest a simple picture of independent spins and small-size clusters of spins flipping in a frozen-random-background field. This picure is similar to the phenomenological description of amorphous materials in terms of two-level systems. Distributions of the quantities which characterize a low-lying energy state in this picture are obtained numerically. A crude analytic calculation of these distributions is also included. These distributions are then used to calculate various low-temperature dynamic properties such as the time-dependent susceptibility, relaxation of the magnetization in an external magnetic field, and the remanent magnetization. We find that this simple description provides qualitative explanations of a large number of results obtained in previous Monte Carlo simulations.

Analysis of ising model critical exponents from high temperature series expansion

Abstract: High temperature series expansion for the critical exponents of the Ising model are reanalysed using a modified ratio method. The analysis shows that a minor modification of the ratio method yields for all lattices a value, for the exponent γ in three dimensions, close to 1.245, therefore lower than the quoted value 1.250, and much closer to the renormalization group (R.G.) value 1.241. The exponent is analysed in two ways : in one method γ is estimated directly while in the other one Tc, is calculated first. With these new values of Tc, the exponent α is recalculated and found to be very close to the R.G. value 0.110. The value of ν is not modified (0.638) and is therefore still a problem for hyperscaling, and is in disagreement with the R.G. value (0.630).

Neutron Scattering Study of a One-Dimensional Ising-Like Antiferromagnet CsCoCl 3 . I. Instantaneous Correlation

Abstract: Instantaneous spin correlations \(\mathscr{S}(\mbi{Q})\) in CsCoCl 3 have been studied by means of quasi-elastic neutron scattering technique and well defined plane-like intensity distribution characteristic for 1D system has been observed. At temperatures higher than 30 K, the observed temperature variation of \(\mathscr{S}(\mbi{Q})\) is in good agreement with the rigorous solution for the 1D Ising antiferromagnet with | J |/ k =75 K. When the temperature is decreased, 3D critical scattering consistent with the previously worked model can be observed at two Neel points T N1 =21.3 K and T N2 =9.2 K.

Ordering and phase transitions in ising systems with competing short range and dipolar interactions

Abstract: As a simple model of order-disorder ferroelectrics or dipolar magnets we consider a simple cubic Ising-system with nearest neighbor exchangeJ and dipolar interaction of strengthµ 2/a 3. ForJa 3/µ 2<−1.3384 the ground-state is antiferromagnetic, while for −1.33840.16429 the ferromagnetic phase becomes stable (with domain arrangements depending on the shape of the sample). For all critical values ofJa 3/µ 2 where the bulk energies of two phases become equal also the interface energy between these phases is found to be zero. The ordering at nonzero temperature is studied by means of mean-field approximations (MFA) and Monte Carlo (MC) calculations. It turns out that forJa 3/µ 2 of order unity the MFA overestimates ordering temperatures by about a factor of two, and predicts multicritical points (between the disordered and two ordered phases) at nonzero temperature, including two biaxial Lifshitz points which the MC work suggests to occur atT=0. In contrast to MFA the layered antiferromagnetic structure is found to be stable only at extremely lowT, because a metastable spin-glass phase (with random arrangement of ferromagnetic rows in the spin direction) has only slightly higher energy. The MFA also yields two regimes of helical phases which are “locked in” to the antiferromagnetic phases at uniaxial Lifshitz points occurring at the Brillouin zone boundary. In the MC-work various methods of treating the long-range interaction are investigated. While all kinds of truncations as well as compensating field methods are rather unsatisfactory in our case, Ewald summation techniques yield satisfactory results. Nevertheless strong fluctuations as well as strong finite size effects prevent us from making accurate exponent estimates, but arguments are given that there is no regime of broad visibility of Landaulike critical behavior. Finally the extension of our results to other lattices as well as experimental applications are briefly discussed.

Variational principle for the distribution function of the effective field for the random Ising model in the Bethe approximation

Abstract: We study the random Ising model in the pair approximation of the cluster variation method. We show that the distribution function of the effective field is determined either by a reducibility condition of the distribution function of two sites to that of one site or by a stationarity condition of the averaged free energy.

Critical properties of site- and bond-diluted Ising ferromagnets

Abstract: A position space renormalisation group technique (decimation) is used to treat site- and bond-diluted Ising ferromagnets on square and triangular lattices. In every case the renormalisation equations lead to two fixed points, one corresponding to the pure Ising system and one to percolation. Linearising around the fixed points, eigenvalues and hence the critical exponents nu T, nu P (correlation length), alpha (specific heat) and phi (crossover) are obtained. Numerical iteration of the recursion relations provides the critical curves. Values of critical temperatures, critical concentrations, exponents, and of limiting slopes of critical curves are in very satisfactory agreement with all known exact or series results. Points on the critical curve flow into the Ising fixed point under the transformation. The sign of the pure Ising exponent alpha is consistent with this flow for the site-diluted cases but not for the bond-diluted cases.

Spin glasses for the infinitely long ranged bond Ising model and for the short ranged binary bond Ising model without use of the replica method

Abstract: The long ranged Gaussian random Ising bond model and short ranged binary random Ising bond model are discussed by the method of the pair approximation of the cluster variation and by the method of the integral equation for the distribution function of the effective fields. For the long ranged model, Sherrington and Kirkpatrick's result is generalized and rederived without use of the replica method. For the short ranged model, i.e. a binary mixture of J A = - J B , the integral equation is solved exactly and the energy of the spin glass state is obtained at T = 0.

The infinite cluster method in the two-dimensional ising model

Abstract: By studying infinite clusters in the two dimensional ferromagnetic Ising model some new results on the problem of existence of non-translation invariant equilibrium states are obtained. Furthermore a new proof of a theorem by Abraham and Reed is given.

Strong Coupling Expansions and Phase Diagrams for the O(2), O(3) and O(4) Heisenberg Spin Systems in Two-dimensions

Abstract: The Ising and $\mathrm{O}(n)$, $2\ensuremath{\le}n\ensuremath{\le}4$, models in two dimensions are studied using a quantum-mechanical Hamiltonian formalism in which a "time" axis is continuous and a spatial axis is discrete. Strong-coupling series for the theory's mass gaps and $\ensuremath{\beta}$ (Callan-Symanzik) functions are computed and are used to search for phase transitions. The critical point and critical index $\ensuremath{ u}$ of the Ising model are found exactly. The critical point of the O(2) model is found (${g}^{*}=1.08$), and the series suggest that the theory's correlation length possesses an essential singularity with the behavior predicted by Kosterlitz. The critical points of the O(3) and O(4) models are predicted to be at zero coupling, i.e., no evidence for a phase transition at nonzero $g$ is found for the non-Abelian models. Interpolating forms (two-point Pad\'e approximants) for these theories' $\ensuremath{\beta}$ (Callan-Symanzik) functions are computed for all $g$. The transition regions between weak and strong coupling are seen to be quite narrow.

Lifshitz points in ising systems

Abstract: Phase diagrams of Ising systems with competing interactions are calculated using (a) the method of Muller-Hartmann and Zittartz to determine the transition temperature via the vanishing of an interface free energy (b) a Migdal-Kadanoff bond-moving scheme and (c) Monte Carlo simulations. It is shown that in two-dimensional Ising systems a uniaxial Lifshitz point can exist at non-zero temperatures, whereas the lower critical dimensiond l for a Lifshitz point in a system with identical competing interactions along each of its cartesian axis isd l ≧2.

Susceptibility and fourth-field derivative of the spin-1/2 Ising model for T>Tc and d=4

Abstract: THe authors investigate the spin-1/2 Ising model with nearest-neighbour interactions on the four-dimensional simple hypercubic lattice. High-temperature series expansions are studied for the zero-field susceptibility chi 0 and the fourth-field derivative of the free energy Xi 0(2) up to order nu 17. The series are analysed for singularities of the form t-1 mod 1nt mod p where t is the reduced temperature. For chi 0 it is found that p=0.33+or-0.07 when q=1, in good agreement with the prediction p=1/3, q=1 of renormalisation group theory. The critical temperature is estimated to be nu c-1=6.7315+or-0.0015. Results for chi 0(2) are more slowly convergent but are not inconsistent with the renormalisation group prediction p=1/3, q=4.

Monte Carlo renormalization-group studies of the d=2 Ising model

Abstract: A Monte Carlo renormalization-group method is described and illustrated by application to the two-dimensional Ising model using several different renormalization-group transformations.

Monte carlo study of an ordering alloy on an fcc lattice

Abstract: We report results of computer simulations of a binary alloy on an fcc lattice, equivalent to an Ising system with a nearest-neighbor antiferromagnetic interaction J > 0 and a next-nearest-neighbor ferromagnetic interaction -..cap alpha..J, ..cap alpha.. > 0. Our data indicate the existence of a discontinous change in energy and in sublattice magnetization as a function of temperature, for small ..cap alpha... For ..cap alpha.. > or approx. = 0.25, the transition appears to be continuous suggesting a tricritical point at some intermediate ..cap alpha...