Ising model in an external field on a hierarchical lattice

The two-dimensional renormalization map of the diamond-hierarchical Ising model in an external field is given, and pictures of the distribution of zeros of the partition function in the complex plane of temperature for varying values of coupling constant and external field are shown. Critical exponents of the model are found, and results are different from those of the Ising model on a two- or three-dimensional regular lattice.     

Square lattice Ising antiferromagnet in an external magnetic field

We study the anisotropic square lattice Ising antiferromagnet in terms of three parameters: an external magnetic field B, an effective temperature and an anisotropy parameterK. The model, i.e., partition function and free energy, is solved exactly in the anisotropic limit,K0, for arbitrary temperature and field by using the transfer matrix method. We also calculate the first corrections beyond this limit. The limit is non trivial and the phase transition is completely preserved. It is of the expected Ising type. The transition temperature _c (B, K) is determined exactly for bothK0 andK and the results are used to check the validity of a recently conjectured formula by Müller-Hartmann and the author.     

Dual properties of lattice systems with broken symmetry: Ising and potts chains in an external field

Dual transformation that produces a path between inhomogeneous spin chains in a site-dependent potential and inhomogeneous spin chains with nonconstant interactions is established. It implies coupling of equivalent random versions of theoretical models. The duality is observed for a number of spin chains with nearest-neighbour as well as multispin interactions.     

One-dimensional Ising model in arbitrary external field

The external potential needed to produce an arbitrary equilibrium density profile for a one-dimensional lattice gas with nearest neighbor interactions is solved exactly. The resulting sequence of direct correlation functions is shown to be of short range, and in the ferromagnetic case the even members alternate in sign at zero spin. The even Ursell distributions in this case likewise alternate in sign.Supported in part by U.S. ERDA under contract E(11-1)-3077.     

The random field method for the Ising model

The random field method is used for investigation of the Ising model. The generalization of the variational principle and a new representation for the free energy of the Ising model are proposed.     

The infinite cluster method in the two-dimensional Ising model

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.This work was supported in part by C.N.R., G.N.F.M.     

Glassy behaviour of random field Ising spins on Bethe lattice in external magnetic field

The thermodynamics and the phase diagram of random field Ising model (RFIM) on Bethe lattice are studied by using a replica trick. This lattice is placed in an external magnetic field (B). A Gaussian distribution of random field (hi) with zero mean and variance hi2 = HR2F is considered. The free-energy (F ), the magnetization (M) and the order parameter (q) are investigated for several values of coordination number (z). The phase diagram shows several interesting behaviours and presents tricritical point at critical temperature TC = J/k and when HRF = 0 for finite z. The free-energy (F) values increase as T increases for different intensities of random field (HRF) and finite z. The internal energy (U) has a similar behaviour to that obtained from the Monte Carlo simulations. The ground state of magnetization decreases as the intensity of random field HRF increases. The ferromagnetic (FM)-paramagnetic (PM) phase boundary is clearly observed only when z →∞. While FM-PM-spin glass (SG) phase boundaries are present for finite z. The magnetic susceptibility (χ) shows a sharp cusp at TC in a small random field for finite z and rounded different peaks on increasing HRF.     

On the mean-field Ising model in a random external field

We use a method developed by van Hemmen to obtain the free energy of the mean-field Ising model in a random external magnetic field. Some results of previous mean-field calculations are confirmed and generalized. The tricritical point in the global phase diagram is discussed in detail. We also consider different probability distributions of the random fields and provide some proofs regarding the conditions for the existence of a tricritical point.     

Generalized transfer formalism and application to random Ising models

We derive an exact formula for the critical temperature of any planar Ising model built up by periodic repetition of an elementary cell. It is shown that the weight of models with analytic free energy increases with increasing size of the elementary cell. As an application we discuss phase diagrams of Ising models with restricted randomness where the coupling arrangement is governed by the probability law of the popular binary distribution.Work performed within the research program of the Sonderforschungsbereich 125 Aachen/Jülich/Köln; also part of: P. Hoever, Dissertation, Köln 1981 (unpublished)     

Glassy phase and thermodynamics for random field Ising model on spherical lattice in magnetic field

Phase diagram and thermodynamic parameters of the random field Ising model (RFIM) on spherical lattice are studied by using mean field theory. This lattice is placed in an external magnetic field (B). The random field (hi) is assumed to be Gaussian distributed with zero mean and a variance     

Dynamic compensation temperature in the kinetic spin-1 Ising model in an oscillating external magnetic field on alternate layers of a hexagonal lattice

The dynamic behavior of a two-sublattice spin-1 Ising model with a crystal-field interaction (D) in the presence of a time-varying magnetic field on a hexagonal lattice is studied by using the Glauber-type stochastic dynamics. The lattice is formed by alternate layers of spins σ=1 and S=1. For this spin arrangement, any spin at one lattice site has two nearest-neighbor spins on the same sublattice, and four on the other sublattice. The intersublattice interaction is antiferromagnetic. We employ the Glauber transition rates to construct the mean-field dynamical equations. Firstly, we study time variations of the average magnetizations in order to find the phases in the system, and the temperature dependence of the average magnetizations in a period, which is also called the dynamic magnetizations, to obtain the dynamic phase transition (DPT) points as well as to characterize the nature (continuous and discontinuous) of transitions. Then, the behavior of the total dynamic magnetization as a function of the temperature is investigated to find the types of the compensation behavior. Dynamic phase diagrams are calculated for both DPT points and dynamic compensation effect. Phase diagrams contain the paramagnetic (p) and antiferromagnetic (af) phases, the p+af and nm+p mixed phases, nm is the non-magnetic phase, and the compensation temperature or the L-type behavior that strongly depend on the interaction parameters. For D<2.835 and H₀>3.8275, H₀ is the magnetic field amplitude, the compensation effect does not appear in the system.     

Interface motion in a two-dimensional Ising model with a field

We determine by Monte Carlo simulations the width of an interface between the stable phase and the metastable phase in a two-dimensional Ising model with a magnetic field, in the case of nonconversed order parameter (Glauber dynamics). At zero temperature, the width increases ast with–1/3, as predicted by earlier theories. As temperature increases, the value of the effective exponent that we measure decreases toward the value 1/4, which is the value in the absence of magnetic field.     

Replica field theory and renormalization group for the Ising spin glass in an external magnetic field

We use the generic replica symmetric cubic field theory to study the transition of short-range Ising spin glasses in a magnetic field around the upper critical dimension. A novel fixed point is found from the application of the renormalization group. In the spin-glass limit, this fixed point governs the critical behavior of a class of systems characterized by a single cubic parameter. For this universality class, the spin-glass susceptibility diverges at criticality, whereas the longitudinal mode remains massive. The third mode, however, behaves unusually. The physical consequences of this unusual behavior are discussed, and a comparison with the conventional de Almeida-Thouless scenario is presented.     

Approximate calculations for the two-dimensional Ising model

A self-consistent molecular field approximation for the two-dimensional, square-lattice Ising model is used to calculate the energy and magnetization. Agreement with the exact calculations is good except near the critical temperature, which differs from the exact critical temperature by 4%. The specific heat has no anomalous behavior asT approachesT _c from above, and the magnetization follows the incorrect Weiss (T _c-T)^1/2 law asT approachesT _c from below.     

Analytical solution of the mean field Ising model for finite systems

The Ising model for finite systems, e.g. for clusters of different sizes and crystal lattices, was solved analytically by the mean field approach. The magnetization was calculated from the number of accessible microstates, using the gamma function and its derivatives, unlike the usual solution in the microcanonical which uses the Stirling approximation. We determined a scaling exponent of ～1/3, which shows how the Curie temperature decreases with decreasing nanoparticle size. Moreover, the model predicts the behaviour of surface and core regions and it explains in simple terms several effects previously observed in experiments and Monte Carlo simulations of small magnetic systems.     

Gauge-invariant lattice gas for the microcanonical Ising model

We introduce a lattice gas for particles with discrete momenta (1, 0, –1) and local deterministic microdynamics, which exactly reproduces Creutz's microcanonical algorithm for the ferromagnetic Ising model. However, because of the manifest gauge invariance of our variables, both the Ising ferromagnetic and spin-glass systems share precisely the same dynamics with different initial conditions. Additional conservation laws in the 1D Ising case result in a completely integrable system in the limit of zero or unbounded demon energy cutoff. Numerical investigations of ergodicity are presented for the pure Ising lattice gas in one and two dimensions.