Resumo em Inglês:

The Ising model is one of the standard models in statistical physics. Since 1969 more than 16,000 publications have appeared using this model. The model was introduced by Ernst Ising in his doctoral dissertation. His life covers most of the past century and was interwoven with striking events in science and politics. It was characterized as a "walk on a tightrope". In this paper some biographical notes and milestones of the early development of the Ising model are given.

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The exact solution of the asymmetric exclusion problem with N distinct classes of particles (c = 1, 2, ..., N), with hierarchical order is presented. In this model the particles (size 1) are located at lattice points, and diffuse with equal asymmetric rates, but particles in a class c do not distinguish those in the classes c' > c from holes (empty sites). We generalize and solve exactly this model by considering the molecules in each distinct class c = 1, 2, ..., N with sizes s c (s c = 0, 1, 2, ...), in units of the lattice spacing. The solution is derived via a Bethe ansatz of nested type.

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We study the spin pair correlation function of the one-dimensional Ising model with competing nearest and next-nearest neighbor interactions, or ANNNI chain, in the presence of an external field. Of particular interest are the disorder lines where exponential decay of the spin pair correlation changes from monotonic to oscillatory. We extend previous studies for higher field values and obtain asymptotic expressions for disorder lines at low temperatures. We also observe reentrant disorder lines.

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This work analyzes the emergence of log-periodic oscillations in thermodynamic functions of Ising models on hierarchical lattices. Several situations, where the exchange interactions are periodic or aperiodic, are taken into account. High precision values for the thermodynamic functions are numerically obtained with the method of transfer matrices. Fitting the curves close to the critical temperature leads to the values of the critical exponents and to the period and amplitude of the oscillations. The first two quantities are found to agree with the results predicted by the renormalization group. The amplitude of oscillations, which are minute for both periodic systems and those with aperiodic irrelevant fluctuations, are significantly enhanced for systems with aperiodic relevant fluctuations. Distinct morphologies of the oscillating pattern are discussed, where oscillations are respectively sinusoidal and with a significant contribution of higher order harmonics.

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In this paper we study damage spreading in a one-dimensional model under two dynamics introduced by Hinrichsen and Domany. In particular, we study the effects of synchronous and asynchronous updating on the spreading properties. We show that the damage does not spread when the second dynamic is implemented in a synchronous way. We find that the rules for updating the damage produced by this dynamics, as the temperature goes to infinity and a certain parameter lambda is zero, are equivalent to those of Grassberger's well-known model A cellular automaton.

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We obtain the local magnetization of a planar Ising model with defects of different types. It is shown that near the critical point the local magnetization has a nonuniversal behavior that manifests itself in the fact that its critical exponent is a continuous function of the microscopic parameters of the system.

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The current status of experiments on the d = 2 and d = 3 random-exchange and random-field Ising models, as realized in dilute anisotropic antiferromagnets, is discussed. Two areas of current investigation are emphasized. For d = 3, the large random field limit is investigated and equilibrium critical behavior is characterized at high magnetic concentrations.

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We study the effect of random bonds and fields on the dynamical behavior of the one-dimensional transverse Ising model with four-spin interactions. We consider finite chains of increasing size to determine the time-dependent correlation function and the longitudinal relaxation function of the infinite chain. In this fully disordered system we observe a crossover from a collective modetype of dynamics to that of a central regime.

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We study the two dimensional quantum Heisenberg antiferromagnet on the square lattice with easy-axis exchange anisotropy by the semiclassical method called pure-quantum self-consistent harmonic approximation. In particular, we focus on the problem of the existence of a nite-temperature transition in such a model, and study the corresponding critical temperature as the spin value and the anisotropy vary. We find that an Ising-like transition characterizes the model even when the anisotropy is of the order of 10-2J (J being the intra-layer exchange integral). The good agreement found between our theoretical results and the experimental data for the compounds Rb2MnF4, K2MnF4, and K2NiF4 shows that the insertion of the easy-axis exchange anisotropy, with quantum effects properly taken into account, provides a quantitative description and explanation of the real system's critical behaviour.

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We discuss the origin of stretched exponential relaxation in disordered Ising spin systems by writing the master equation on the phase space, and the evolution of local and global spin autocorrelation functions, in terms of independent relaxation modes, which are eigenvectors of the time evolution operator. In this sense it is shown that when the relaxation modes are spatially delocalized, both local and global autocorrelation functions may present non-exponential relaxation. We also analyze results for random walks on the dilute hypercube, which may be associated with the phase space of a disordered Ising spin system. As expected, the results show a stretched exponential relaxation near the percolation transition, since it deals with random walks on a fractal percolating cluster defined on a closed surface. We argue that the same type of topology is present in the available region of configuration space in Ising spin-glass systems near the glass transition, since these systems present very similar relaxation patterns in this temperature range.

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We propose an improvement of a Monte Carlo method designed to treat the Ising model in a field [C. Lieu and J. Florencio, J. Low Temp. Phys. 89, 565 (1992)]. The method involves the counting of bonds linking neighboring like-spins and yields the degeneracy of the system's energy states, hence the partition function. There is no acceptance-rejection procedure and all the randomly generated configurations are kept. The sampling depends on geometry only, so results of a given run can be used for all temperatures and energy parameters. In order to understand the virtues and inadequacies of the method, we obtained exact results for small lattices. We find that a Monte Carlo run must be followed by a Gaussian fit in order to account properly for the rare events not recorded in the sampling. Finally, we also established bounds for the location of the peak for the specifc heat of the Ising model in a magnetic field in two dimensions for several values of the field in the thermodynamic limit.

Resumo em Inglês:

The name Ising has come to stand not only for a specific model, but for an entire universality class - arguably the most important such class - in the theory of critical phenomena. I review several examples, both in and out of equilibrium, in which Ising universality appears or is pertinent. The "Ornstein-Zernike" connection concerns a thermodynamically self-consistent closure of the eponymous relation, which lies at the basis of the modern theory of liquids, as applied to the Ising lattice gas. Debye and Hückel founded the statistical mechanics of ionic solutions, which, despite the long-range nature of the interaction, now appear to exhibit Ising-like criticality. The model of Widom and Rowlinson involves only excluded-volume interactions between unlike species, but again belongs to the Ising universality class. Far-from-equilibrium models of voting behavior, catalysis, and hysteresis provide further examples of this ubiquitous universality class.

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Ground-state and thermodynamic properties of the spin-1/2 two-dimensional easy-plane XXZ model are investigated by both a Green's-function approach and by Lanczos diagonalization on lattices with up to 36 sites. We calculate the spatial and temperature dependences of various spin correlation functions, as well as the wave-vector dependence of the spin susceptibility for all anisotropy parameters delta. In the easy plane ferromagnetic region (-1 < delta < 0), the longitudinal correlators of spins at distance r change sign at a nite temperature T0 (delta, r). This transition, observed in the 2D case for the first time, can be interpreted as a quantum to classical crossover.

Resumo em Inglês:

The Ising model is known widely for studying equilibrium behavior. We show that the model is also useful for studying nonequilibrium behavior in some special situations. The method of recurrence relations has been applied to obtain the time evolution of a non-commuting spin operator. Also obtained are the structure functions and their time dependent behavior. It is shown that the transverse component of the static susceptibility can be obtained from the dynamic results.

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The kinetic Ising model on an n-isotopic chain is considered in the framework of Glauber dynamics. The chain is composed of N segments with n sites, each one occupied by a different isotope. Due to the isotopic mass difference, the n spins in each segment have different relaxation times in the absence of interactions, and consequently the dynamics of the system is governed by multiple relaxation mechanisms. The solution is obtained in closed form for arbitrary n, by reducing the problem to a set of n coupled equations, and it is shown rigorously that the critical exponent z is equal to 2. Explicit results are obtained numerically for any temperature and it is also shown that the dynamic susceptibility satisfies the new scaling (Nagel scaling) proposed for glass-forming liquids. This is in agreement with our recent results (L. L. Gonçalves, M. López de Haro, J. Tagüeña-Martínez and R. B. Stinchcombe, Phys. Rev. Lett. 84 , 1507 (2000)), which relate this new scaling function to multiple relaxation processes.

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Square water takes into account the directionality of hydrogen bonds. The model is reviewed and its properties as a solvent for apolar particles are studied through Monte Carlo simulations. Specific heat measurements are used to identify phase separation. Data for comparison with the lattice gas on the square lattice are presented and the relation to non-associating solvents is discussed. Data for the frequency of hydrogen bonds as a function of temperature indicate a slower rate of bond breaking for the hydration shell as compared to bulk water particles.

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We review recent investigations of the critical behavior of ferromagnetic q-state Potts models on a class of hierarchical lattices, with exchange interactions according to some deterministic but aperiodic substitution rules. The problem is formulated in terms of exact recursion relations on a suitable parameter space. The analysis of the fixed points of these equations leads to a criterion to gauge the relevance of the aperiodic geometric fluctuations. For irrelevant fluctuations, the critical behavior remains unchanged with respect to the underlying uniform models. In the presence of relevant fluctuations, a non-trivial symmetric fixed point, associated with the critical behavior of the uniform model, becomes fully unstable, and there appears a two-cycle of the recursion relations. A scaling analysis, supported by direct numerical thermodynamical calculations, shows the existence of a novel critical universality class associated with relevant geometric fluctuations.

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The instability of the paraelectric phase in ferroelectrics, driven by thermal fluctuations, is discussed on the basis of the quantum three-dimensional spin-1/2 transverse-field Ising model (TIM) within the framework of the Green function method. The two-step critical dynamics of the TIM is analyzed through the ferroelectric order-parameter fluctuation spectra observed above the critical temperature Tc. The spectra profiles are given near Tc in explicit form. The slow-down exponents deltac = <img src="http:/img/fbpe/bjp/v30n4/19eq01.gif" align="absmiddle"> and deltas = <img src="http:/img/fbpe/bjp/v30n4/19eq02.gif" align="absmiddle"> are deduced for the slow and fast parts of structural relaxation and are compared with those known from the literature.

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Computer simulations have played an important role in the elucidation of wetting and interface unbinding phenomena. In particular, use of the Ising-lattice-gas model in a film geometry and subject to diverse surface and bulk magnetic fields has permitted extensive Monte Carlo simulations to reveal new features of the phase diagrams associated with these phenomena and to provoke new theoretical studies. The status of our knowledge about the nature of wetting and interface-delocalization transitions which has resulted from these Ising model simulations will be summarized.

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We study the Ising axis conversion in a mixed Tb0.5Dy0.5Cu2 single crystal. Interest is focused on how changes in the exchange interactions due to rare earth substitutions influence the existence of magnetic phases and the critical field values for the Ising axis conversion. From magnetisation measurements we determined the (H - T ) phase diagram for magnetic fields parallel to the easy a-axis and the temperature dependence of the critical field for the Ising axis conversion. Both properties for the mixed crystal follow a simple composition scaling behavior. But in contrast to previous studies on the pure compounds TbCu2 and DyCu2 the changes of magnetic and structural properties at the conversion cannot be recovered completely by thermal treatment. Only a small part (10 % of the sample volume) goes back to the virgin state after warming the sample to 500 K. This behavior is of great interest for further neutron or X-ray diffraction studies of the Ising axis conversion allowing to study the converted phase under routinely used experimental conditions.

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Instabilities of long-range order are observed in samples of the d = 3 dilute uniaxial antiferromagnet Fe xZn1-xF2 with x = 0:56 and 0.41, under strong random fields. The onset of instability, mapped in the (H, T) phase diagrams of the samples, reveals that the H, T and x dependence of the effective random field is in qualitative agreement with a mean-field expression predicted in the weak-field limit for site diluted antiferromagnets.

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Experimentally derived physical properties of 3d Ising spin glasses are discussed and compared to what is anticipated from theory and simulations of model systems. It is found that the spin glass experiences a zero-field phase transition to a low-temperature spin-glass phase, that there is no phase transition in a magnetic field and that nonequilibrium and chaos effects often dominate the dynamics of the low-temperature glassy phase. These results are in agreement with findings from Monte Carlo simulations and with properties derived from the droplet model.

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The Broad Histogram Method (BHM) allows one to determine the energy degeneracy g(E), i.e. the energy spectrum of a given system, from the knowledge of the microcanonical averages < Nup (E) > and < Ndn (E) > of two macroscopic quantities Nup and Ndn defined within the method. The fundamental BHM equation relating g(E) to the quoted averages is exact and completely general for any conceivable system. Thus, the only possible source of numerical inaccuracies resides on the measurement of the averages themselves. In this text, we introduce a Monte Carlo recipe to measure microcanonical averages. In order to test its performance, we applied it to the Ising ferromagnet on a 32 × 32 square lattice. The exact values of g(E) are known up to this lattice size, thus it is a good standard to compare our numerical results with. Measuring the deviations relative to the exactly known values, we verified a decay proportional to 1/<img src="http:/img/fbpe/bjp/v30n4/24eq01.gif" border="0">, by increasing the counter (counts) of averaged samples over at least 6 decades. That is why we believe this microcanonical simulator presents no bias besides the normal statistical fluctuations. For counts ~ 10(10), we measured relative deviations near 10(5) for both g(E) and the specific heat peak, obtained through BHM relation.

Resumo em Inglês:

Recent studies of the effects of geometric fluctuations, associated with aperiodic exchange inter-actions, on the critical behavior of ferromagnetic Ising models on hierarchical lattices, were the motivation to investigate some properties of two-letter sequences generated by uniform substitutions. We rigorously identify the substitutions that generate either periodic or aperiodic (almost periodic or not) sequences. For instance, two known substitutions, the Thue-Morse and the period-doubling rules, generate aperiodic sequences of almost periodic type.

Resumo em Inglês:

The Ising model has been an important theoretical tool in the understanding of phase transitions in ferroelectric materials. We first review how it relates to the underlying physics of order-disorder phase transitions in these systems, as well as mean-field results for the spontaneous polarization and the dielectric constant near the critical temperature. For hydrogen-bonded ferroelectrics, a term of interaction with an external transverse field is necessary to account for proton tunneling between the two minima of a double-well potential. Finally, we discuss both experimental and theoretical results for the problems of proton-lattice interactions, central peak dynamics, dynamical behavior of pseudo-one-dimensional ferroelectrics and pressure effects on hydrogen bonded ferroelectrics.

Resumo em Inglês:

We discuss order-disorder phase transitions of classical and quantum Ising models on the basis of the Ginzburg-Landau-Wilson free-energy action and a modification of a scaling method proposed by Thompson. The two-time critical relaxation scenario, driven by thermal and quantum fluctuations, respectively, is given in terms of the slow-down critical exponent of the order-parameter excitations.

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Monte Carlo simulations with up to 176(5) spins confirm the Chen-Dohm theory for the five-dimensional Ising model. Also world record sizes 1000192²; 9984³; 880(4); 48(6) and 21(7) spins were simulated in the literature, and we describe the needed multi-spin coding algorithm.

Resumo em Inglês:

The factors that make certain magnetic materials behave similarly to corresponding Ising models are reviewed. Examples of extensively studied materials include Dy(C2H5SO4)3.9H2) (DyES), Dy3Al5O12 (DyAlG), DyPO4, Dy2Ti2O7, LiTbF4, K2CoF4, and Rb2CoF4. Various comparisons between theory and experiment for these materials are examined. The agreement is found to be generally very good, even when there are clear differences between the ideal Ising model and the real materials. In a number of experiments behavior has been observed that requires extensions of the usual Ising model. These include the effects of long range magnetic dipole interactions, competing interaction effects in field-induced phase transitions, induced staggered field effects and frustration effects, and dynamic effects. The results show that the Ising model and real magnetic materials have provided an unusually rich and productive field for the interaction between theory and experiment over the past 40 years.