Showing papers on "Ising model published in 1996"

Quantum Ising Phases and Transitions in Transverse Ising Models

Abstract: Introduction -- Transverse Ising Chain (Pure System) -- Transverse Ising System in Higher Dimensions (Pure Systems) -- ANNNI Model in Transverse Field -- Dilute and Random Transverse Ising Systems -- Transverse Ising Spin Glass and Random Field Systems -- Dynamics of Quantum Ising Systems -- Quantum Annealing -- Applications -- Related Models -- Brief Summary and Outlook -- Index.

Quantum critical behavior for a model magnet

Abstract: The classical, thermally driven transition in the dipolar-coupled Ising ferromagnet LiHoF_4 (T_c=1.53K) can be converted into a quantum transition driven by a transverse magnetic field H_t at T=0. The transverse field, applied perpendicular to the Ising axis, introduces channels for quantum relaxation, thereby depressing T_c. We have determined the phase diagram in the H_t−T plane via magnetic susceptibility measurements. The critical exponent, γ=1, has a mean-field value in both the classical and quantum limits. A solution of the full mean-field Hamiltonian using the known LiHoF_4 crystal-field wave functions, including nuclear hyperfine terms, accurately matches experiment.

Theory of Branching and Annihilating Random Walks

Abstract: Nonequilibrium models with an extensive number of degrees of freedom whose dynamics violates detailed balance occur in studies of many biological, chemical and physical systems. Like equilibrium systems, their stationary states may exhibit phase transitions which in many cases appear to fall into distinct classes characterized by universal quantities such as critical exponents. One of the most common such classes is that exemplified by directed percolation (DP) [1]. This represents a transition from a nontrivial ‘active’ steady state to an absorbing ‘inactive’ state with no fluctuations. Many nonequilibrium phase transitions appear to belong to this universality class, e.g., the contact process [2], the dimer poisoning problem in the ZGB model [3], and auto–catalytic reaction models [4]. The universal properties of the DP transition are theoretically well understood in the context of a renormalization group (RG) analysis based on an expansion around mean field theory below the upper critical dimension dc = 4 [5]. More recently a class of models has been studied which, in certain cases, appear as exceptions to the general rule that such transitions should fall into the DP universality class. These include a probabilistic cellular automaton model [6], certain kinetic Ising models [7,8], and an interacting monomer–dimer model [9]. In one dimension the dynamics of these is equivalent to a class of models called branching and annihilating random walks (BARWs) [10–12], which also have a natural generalization to higher dimensions. In the language of reaction– diffusion systems, BARWs describe the stochastic dynamics of a single species of particles A undergoing three basic processes: diffusion, often modeled by a random walk on a lattice and characterized by a diffusion coefficient D; an annihilation reaction A + A → ⊘ when particles are close (or on the same site), at rate λ; and a branching process A → (m + 1)A (where m is a positive integer), at rate σm. The above–mentioned one– dimensional models all correspond to the case m = 2. For the kinetic Ising model, the particles A are to be identified with the domain walls, and the transition to the inactive state corresponds to the ordering of the Ising spins [7,8]. In general, this new universality class has been observed in d = 1 for even values of m, when the number of particles is locally conserved modulo 2. When m is odd, the DP values of the exponents appear to be realized. (It should be remarked that several of the models which have been studied do not contain three independent parameters corresponding to D, λ, and σm so that it may occur that the actual transition is inaccessible. This appears to be so for the simplest lattice BARW model with m = 2, which is always in the inactive phase [10].) Besides the appearance of a new universality class, another issue which clearly requires theoretical explanation is the occurrence of a transition at a finite value of σm. For the mean field rate equation for the average density u n(t) = −2λ n(t) 2 + mσm n(t) (1)

Surface order large deviations for Ising, Potts and percolation models

Abstract: We derive uniform surface order large deviation estimates for the block magnetization in finite volume Ising (or Potts) models with plus or free (or a combination of both) boundary conditions in the phase coexistence regime ford≧3. The results are valid up to a limit of slab-thresholds, conjectured to agree with the critical temperature. Our arguments are based on the renormalization of the random cluster model withq≧1 andd≧3, and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability. The results are new even for the case of independent percolation (q=1). As a byproduct of our methods, we obtain further results in the FK model concerning semicontinuity (inp andq) of the percolation probability, the second largest cluster in a box and the tail of the finite cluster size distribution.

Hysteresis, avalanches, and disorder-induced critical scaling: A renormalization-group approach

Abstract: Hysteresis loops are often seen in experiments at first-order phase transformations, when the system goes out of equilibrium. They may have a macroscopic jump (roughly as in the supercooling of liquids) or they may be smoothly varying (as seen in most magnets). We have studied the nonequilibrium zero-temperature random-field Ising-model as a model for hysteretic behavior at first-order phase transformations. As disorder is added, one finds a transition where the jump in the magnetization (corresponding to an infinite avalanche) decreases to zero. At this transition we find a diverging length scale, power-law distributions of noise (avalanches), and universal behavior. We expand the critical exponents about mean-field theory in 6-\ensuremath{\epsilon} dimensions. Using a mapping to the pure Ising model, we Borel sum the 6-\ensuremath{\epsilon} expansion to O(${\mathrm{\ensuremath{\epsilon}}}^{5}$) for the correlation length exponent. We have developed a method for directly calculating avalanche distribution exponents, which we perform to O(\ensuremath{\epsilon}). Our analytical predictions agree with numerical exponents in two, three, four, and five dimensions [Perkovi\ifmmode \acute{c}\else \'{c}\fi{} et al., Phys. Rev. Lett. 75, 4528 (1995)]. \textcopyright{} 1996 The American Physical Society.

Exact distribution of energies in the two-dimensional ising model.

Abstract: The low-temperature series expansion for the partition function of the two-dimensional Ising model on a square lattice can be determined exactly for finite lattices using Kaufman's generalization of Onsager's solution. The exact distribution function for the energy can then be determined from the coefficients of the partition function. This provides an exact solution with which one can compare energy histograms determined in Monte Carlo simulations. This solution should prove useful for detailed studies of statistical and systematic errors in histogram reweighting.

Boundary conformal field theory approach to the two-dimensional critical Ising model with a defect line

Abstract: We study the critical two-dimensional Ising model with a defect line (altered bond strength along a line) in the continuum limit. By folding the system at the defect line, the problem is mapped to a special case of the critical Ashkin-Teller model, the continuum limit of which is the $Z_2$ orbifold of the free boson, with a boundary. Possible boundary states on the $Z_2$ orbifold theory are explored, and a special case is applied to the Ising defect problem. We find the complete spectrum of boundary operators, exact two-point correlation functions and the universal term in the free energy of the defect line for arbitrary strength of the defect. We also find a new universality class of defect lines. It is conjectured that we have found all the possible universality classes of defect lines in the Ising model. Relative stabilities among the defect universality classes are discussed.

The magnetization of the 3D Ising model

Abstract: We present highly accurate Monte Carlo results for simple cubic Ising lattices containing up to spins. These results were obtained by means of the Cluster Processor, a newly built special-purpose computer for the Wolff cluster simulation of the 3D Ising model. We find that the spontaneous magnetization M(t) is accurately described by , where , in a wide temperature range 0.0005 < t < 0.26. Any corrections to scaling with higher powers of t could not be resolved from our data, which implies that they are very small. The magnetization exponent is determined as . An analysis of the magnetization distribution near criticality yields a new determination of the critical point: , with a standard deviation of .

3D Ising Model:The Scaling Equation of State

Abstract: The equation of state of the universality class of the 3D Ising model is determined numerically in the critical domain from quantum field theory and renormalization group techniques. The starting point is the five loop perturbative expansion of the effective potential (or free energy) in the framework of renormalized $\phi^4_3$ field theory. The 3D perturbative expansion is summed, using a Borel transformation and a mapping based on large order behaviour results. It is known that the equation of state has parametric representations which incorporate in a simple way its scaling and regularity properties. We show that such a representation can be used to accurately determine it from the knowledge of the few first coefficients of the expansion for small magnetization. Revised values of amplitude ratios are deduced. Finally we compare the 3D values with the results obtained by the same method from the $\epsilon=4-d$ expansion.

Global Persistence Exponent for Nonequilibrium Critical Dynamics.

Abstract: probability, p(t) � t � , that the global order parameter has not changed sign in the time interval t following a quench to the critical point from a disordered state. This exponent is calculated in mean-field theory, in the n = 1 limit of the O(n) model, to first order in ǫ = 4 d, and for the 1-d Ising model. Numerical results are obtained for the 2-d Ising model. We argue that θ is a new independent exponent. For many years it was believed that critical phenomena were characterized by a set of three critical exponents, comprising two independent static exponents (other static exponents being related to these by scaling laws) and the dynamical exponent z. Then, quite recently, it was discovered that there is another dynamical exponent, the ‘non-equilibrium’ (or ‘short-time’) exponent λ, needed to describe two-time correlations in a system relaxing to the critical state from a disordered initial condition [1,2]. It is natural to ask ‘Are there any more independent critical exponents?’. In this Letter we propose such an exponent – the ‘persistence exponent’ θ associated with the probability, p(t) ∼ t −θ , that the global order parameter has not changed sign in time t following a quench to the critical point. We calculate θ in mean-field theory, in the n = ∞ limit of the O(n) model, to first order in ǫ = 4−d (d = dimension of space) and for the d = 1 Ising model. In fact, it turns out that all these results satisfy the scaling law θz = λ−d+1−η/2, which can be derived on the assumption that the dynamics of the global order parameter is a Markov process. We shall argue, however, that this process is in general non-Markovian, so that θ is in general a new, non-trivial critical exponent. The persistence exponent θ was first introduced in the context of the non-equilibrium coarsening dynamics of systems at zero temperature [3,4]. In that context it describes the power-law decay, p(t) ∼ t −θ , of the probability that the local order parameter φ(x) has not changed sign during the time interval t after the quench to T = 0. Equivalently, it gives the fraction of space in which the order parameter has not changed sign up to time t. More generally, one can consider the probability p0(t1, t2) of no sign changes between t1 and t2. Scaling considerations suggest p0(t1, t2) = f(t1/t2) ∼ (t1/t2) θ for t2 ≫ t1. Exact solutions for one-dimensional systems [4,5] indicate that, in general, θ is a new non-trivial exponent for coarsening dynamics. Recently, we have shown that even the diffusion equation exhibits a nontrivial persistence exponent, and have developed a rather accurate approximate theory for this case [6]. The diffusion equation is itself a model of ordering dynamics, via the approximate theory of Ohta, Jasnow and Kawasaki (OJK) [7], and also describes, in its essential features, the ordering kinetics of the nonconserved O(n) model in the large-n limit [8]: The exponents θ for these systems (OJK and large-n) are just those of the diffusion equation. In this Letter we introduce and calculate the analogous exponent θ for non-equilibrium critical dynamics. In this case however, one needs to consider the global, rather than the local order parameter. This is because individual degrees of freedom (‘spins’, say) are rapidly flipping so that the probability of not flipping in an interval t has an exponential tail. We shall see, however, that the probability for the global order parameter not to have flipped indeed decays as a power

On the extremality of the disordered state for the Ising model on the Bethe lattice

Abstract: We give a simple proof that the limit Ising Gibbs measure with free boundary conditions on the Bethe lattice with the forward branching ratio k≥2 is extremal if and only if β is less or equal to the spin glass transition value, given by tanh(β c SG = 1/√k.

Gibbs-Thomson formula for small island sizes: Corrections for high vapor densities.

Abstract: In this paper we report simulation studies of equilibrium features, namely, circular islands on model surfaces, using Monte Carlo methods. In particular, we are interested in studying the relationship between the density of vapor around a curved island and its curvature. The ``classical'' form of this relationship is the Gibbs-Thomson formula, which assumes that the vapor surrounding the island is an ideal gas. Numerical simulations of a lattice gas model, performed for various sizes of islands, do not fit very well to the Gibbs-Thomson formula. We show how corrections to this form arise at high vapor densities, wherein a knowledge of the exact equation of state (as opposed to the ideal-gas approximation) is necessary to predict this relationship. By exploiting a mapping of the lattice gas to the Ising model, one can compute the corrections to the Gibbs-Thomson formula using high field series expansions. The corrected Gibbs-Thomson formula matches very well with the Monte Carlo data. We also investigate finite size effects on the stability of the islands both theoretically and through simulations. Finally, the simulations are used to study the microscopic origins of the Gibbs-Thomson formula. It is found that smaller islands have a greater adatom detachment rate per unit length of island perimeter. This is principally due to a lower coordination of edge atoms and a greater availability of detachment moves relative to edge moves. A heuristic argument is suggested in which these effects are partially attributed to geometric constraints on the island edge. \textcopyright{} 1996 The American Physical Society.

Quantum paraelectric and induced ferroelectric states in

Abstract: Nominally pure has been studied by dielectric spectroscopy using small (linear regime) as well as large electrical fields (non-linear regime) up to . In addition measurements of the specific heat and its field-dependent contribution have been carried out. The field dependence of the dielectric constant and the specific heat can be well described by the transverse Ising Hamiltonian including tunnelling and external field terms. It gives evidence for the existence of polar clusters at low temperatures which are supposed to be associated with the quantum paraelectric state below in accord with recent free-energy calculations. The low-field third-harmonic susceptibility which measures the polar correlations exhibits anomalies near . At high fields an aligned domain state is induced. These results as well as those on the remanent polarization and the dielectric loss have allowed us to deduce a complex E, T-phase diagram. The assignment of the various phases is discussed in connection with the recent proposal of the appearance of a macroscopic quantum state.

Chain length dependence of the polymer-solvent critical point parameters

Abstract: We report grand canonical Monte Carlo simulations of the critical point properties of homopolymers within the bond fluctuation model. By employing configurational bias Monte Carlo methods, chain lengths of up to N=60 monomers could be studied. For each chain length investigated, the critical point parameters were determined by matching the ordering operator distribution function to its universal fixed‐point Ising form. Histogram reweighting methods were employed to increase the efficiency of this procedure. The results indicate that the scaling of the critical temperature with chain length is relatively well described by Flory theory, i.e., Θ−Tc∼N−0.5. The critical volume fraction, on the other hand, was found to scale like φc∼N−0.37, in clear disagreement with the Flory theory prediction φc∼N−0.5, but in good agreement with experiment. Measurements of the chain length dependence of the end‐to‐end distance indicate that the chains are not collapsed at the critical point.

On discrete three-dimensional equations associated with the local Yang-Baxter relation

Abstract: The local Yang-Baxter equation (YBE), introduced by Maillet and Nijhoff, is a proper generalization to three dimensions of the zero curvature relation. Recently, Korepanov has constructed an infinite set of integrable three-dimensional lattice models, and has related them to solutions to the local YBE. The simplest Korepanov model is related to the star-triangle relation in the Ising model. In this Letter the corresponding discrete equation is derived. In the continuous limit it leads to a differential three-dimensional equation, which is symmetric with respect to all permutations of the three coordinates. A similar analysis of the star-triangle transformation in electric networks leads to the discrete bilinear equation of Miwa, associated with the BKP hierarchy.

Surface Tension in Ising Systems with Kac Potentials

Abstract: , d¸2, and fix the temperature below its Lebowitz-Penrose critical value. We prove that when the Kac scaling parameter ° vanishes, the log of the probability of an interface becomes proportional to its area and the surface tension, related to the proportionality constant, converges to the van der Waals surface tension. The results are based on the analysis of the rate functionals for Gibbsian large deviations and on the proof that they i-converge to the perimeter functional of geometric measure theory (which extends the notion of area). Our consider- ations include non smooth interfaces proving that the Gibbsian probability of an interface depends only on its area and not on its regularity.

Two-dimensional planar ferromagnetic coupling in LaMnO3

Abstract: Neutron-scattering experiments have been performed on a LaMnO 3 single crystal. The spin waves exhibit two-dimensional anisotropic dispersion, that is, strong planar ferromagnetic coupling and weak antiferromagnetic interplane coupling with a finite gap. The critical exponent β of the order parameter is 0.25 ±0.01, which is much smaller than values obtained for the three-dimensional Ising (0.326) and Heisenberg (0.367) models. Ferromagnetic spin-wave-like dispersion remains for ω> 10 meV at T c ≈140 K, though the spectrum near the zone center is entirely overdamped around ω= 0.

Survival Probability of a Gaussian Non-Markovian Process: Application to the T=0 Dynamics of the Ising Model.

Abstract: We study the decay of the probability for a non-Markovian stationary Gaussian walker not to cross the origin up to time $t$. This result is then used to evaluate the fraction of spins that do not flip up to time $t$ in the zero temperature Monte Carlo spin flip dynamics of the Ising model. Our results are compared to extensive numerical simulations.

Chain length dependence of the polymer-solvent critical point parameters

Abstract: We report grand canonical Monte Carlo simulations of the critical point properties of homopolymers within the Bond Fluctuation model. By employing Configurational Bias Monte Carlo methods, chain lengths of up to N=60 monomers could be studied. For each chain length investigated, the critical point parameters were determined by matching the ordering operator distribution function to its universal fixed-point Ising form. Histogram reweighting methods were employed to increase the efficiency of this procedure. The results indicate that the scaling of the critical temperature with chain length is relatively well described by Flory theory, i.e. \Theta-T_c\sim N^{-0.5}. The critical volume fraction, on the other hand, was found to scale like \phi_c\sim N^{-0.37}, in clear disagreement with the Flory theory prediction \phi_c\sim N^{-0.5}, but in good agreement with experiment. Measurements of the chain length dependence of the end-to-end distance indicate that the chains are not collapsed at the critical point.

Broad Histogram Method

Abstract: Ferrenberg and Swendsen histogram method is based on Boltzmann probability distribution which presents exponentially decaying tails. Thus, it gives accurate measures only within a narrow window around the simulated temperature. The larger the system, the narrower this window, and the worst the performance of this method. We present a quite different approach, defining a non-biased random walk along the E axis with long range power-law decaying tails, and measuring directly the degeneracy g(E), without thermodynamic constraints. Our arguments are general (model independent), and the method is shown to be exact for the 1D Ising ferromagnet. Also for the 2D Ising ferromagnet, our numerical results for different thermodynamic quantities agree quite well with exact expressions.

Quantum paraelectric and induced ferroelectric states in SrTiO3

Abstract: Nominally pure SrTiO3 has been studied by dielectric spectroscopy using small (linear regime) as well as large electrical fields (non-linear regime) up to 1 kV mm−1. In addition measurements of the specific heat and its field-dependent contribution have been carried out. The field dependence of the dielectric constant and the specific heat can be well described by the transverse Ising Hamiltonian including tunnelling and external field terms. It gives evidence for the existence of polar clusters at low temperatures which are supposed to be associated with the quantum paraelectric state below Tq ≈ 37 K in accord with recent free-energy calculations. The low-field third-harmonic susceptibility which measures the polar correlations exhibits anomalies near Tq . At high fields an aligned domain state is induced. These results as well as those on the remanent polarization and the dielectric loss have allowed us to deduce a complex E, T phase diagram. The assignment of the various phases is discussed in connection with the recent proposal of the appearance of a macroscopic quantum state.

Accuracy of the density-matrix renormalization-group method

Abstract: White's density-matrix renormalization-group (DMRG) method has been applied to the one-dimensional Ising model in a transverse field (ITF), in order to study the accuracy of the numerical algorithm. Due to the exact solubility of the ITF for any finite chain length, the errors introduced by the basis truncation procedure could have been directly analyzed. By computing different properties, like the energies of the low-lying levels or the ground-state one- and two-point correlation functions, we obtained a detailed picture of how these errors behave as functions of the various model and algorithm parameters. Our experience with the ITF contributes to a better understanding of the DMRG method, and may facilitate its optimization in other applications.

Defect Lines in the Ising Model and Boundary States on Orbifolds.

Abstract: Critical phenomena in the two-dimensional Ising model with a defect line are studied using boundary conformal field theory on the $c\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ orbifold. Novel features of the boundary states arising from the orbifold structure, including continuously varying boundary critical exponents, are elucidated. New features of the Ising defect problem are obtained including a novel universality class of defect lines and the universal boundary to bulk crossover of the spin correlation function.

Boundary s matrix for the tricritical ising model

Abstract: The tricritical Ising model perturbed by the subleading energy operator was known to be an integrable scattering theory of massive kinks,14 and in fact it preserves supersymmetry. We consider here the model defined on the half-plane with a boundary and compute the associated factorizable boundary S matrix. The conformal boundary conditions of this model are identified and the corresponding S matrices are found. We also show how some of these S matrices can be perturbed and generate “flows” between different boundary conditions.

Metastability of the Three Dimensional Ising Model on a Torus at Very Low Temperatures

Abstract: We study the metastability of the stochastic three dimensional Ising model on a finite torus under a small positive magnetic field at very low temperatures.