Showing papers on "Ising model published in 2003"
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01 Jan 2003
TL;DR: In this paper, the capacity of the perceptron was investigated using the Hopfield model and the p-Spin Interaction Model at low temperature, and the Diluted SK Model and the K-Sat Problems.
Abstract: 0. Introduction.- 1. A Toy Model, the REM.- 2. The Sherrington-Kirkpatrick Model.- 3. The Capacity of the Perceptron: The Ising Case.- 4. Capacity of the Perceptron: The Gaussian and the Spherical Case.- 5. The Hopfield Model.- 6. The p-Spin Interaction Model at Low Temperature.- 7. The Diluted SK Model and the K-Sat Problems.- 8. An Assignment Problem.- A. Appendix.- Elements of Probability Theory.- References.- Index.
291 citations
TL;DR: In this article, the authors studied the ±J random-plaquette Z-2 gauge model (RPGM) in three dimensions, a three-dimensional analog of the two-dimensional random-bonding Ising model (RBIM), and measured the critical concentration p_c of wrong-sign plaquettes at the confinement-Higgs phase transition along the Nishimori line.
257 citations
TL;DR: In this paper, the Sznajd model is treated analytically on a complete graph and the existence of the phase transition in the original formulation of the model is proved, while for the Ochrombel modification they find smooth behaviour without transition.
Abstract: The Sznajd model, which describes opinion formation and social influence, is treated analytically on a complete graph. We prove the existence of the phase transition in the original formulation of the model, while for the Ochrombel modification we find smooth behaviour without transition. We calculate the average time to reach the stationary state as well as the exponential tail of its probability distribution. An analytical argument for the observed 1/n dependence in the distribution of votes in Brazilian elections is provided.
193 citations
TL;DR: In this article, it was shown that nonperturburburbative renormalization group equations allow one to obtain very accurate critical exponents for the three-dimensional Ising model.
Abstract: On the example of the three-dimensional Ising model, we show that nonperturbative renormalization group equations allow one to obtain very accurate critical exponents. Implementing the order ${\ensuremath{\partial}}^{4}$ of the derivative expansion leads to $\ensuremath{
u}=0.632$ and to an anomalous dimension $\ensuremath{\eta}=0.033$ which is significantly improved compared with lower orders calculations.
165 citations
TL;DR: Under the best conditions the simulated output of a coupled lattice of receptors closely matched the time-course and amplitude found experimentally in living bacteria, and shows an unexpected emergence of spatial patterns of methylation within the receptor lattice.
161 citations
TL;DR: In this article, the authors examined the eigenstate of the Heisenberg spin Hamiltonian and the Ising spin Hamiltonians for a general spin dimer consisting of M unpaired spins at one spin site and N unpaired spin sites at the other spin site.
Abstract: We examined the eigenstates of the Heisenberg spin Hamiltonian Ĥ=−JŜ1⋅Ŝ2 and the Ising spin Hamiltonian ĤIsing=−JŜ1zŜ2z for a general spin dimer consisting of M unpaired spins at one spin site and N unpaired spins at the other spin site, and then analyzed how the broken-symmetry spin state of a spin dimer is related to the eigenstates of Ĥ and ĤIsing. Our work shows that the description of the highest-spin and broken-symmetry spin states of a spin dimer by Ĥ is the same as that by ĤIsing. For the analysis of spin exchange interactions of a magnetic solid on the basis of density functional theory, the use of the Heisenberg spin Hamiltonian in the “cluster” approach is consistent with that of the Ising spin Hamiltonian in the “noncluster” approach.
147 citations
TL;DR: In this paper, a review of Monte Carlo simulations of ferromagnetic Ising lattices in a film geometry with surface magnetic fields is presented, and the authors summarize predictions of scaling theories.
Abstract: We present a brief review of Monte Carlo simulations of ferromagnetic Ising lattices in a film geometry with surface magnetic fields. The seminal work of Nakanishi and Fisher [Phys. Rev. Lett.
49:1565 (1982)] showed how phase transitions in such models are related to wetting in systems with short range forces; and we will show how theoretical concepts about critical and tricritical wetting, interface localization-delocalization, and capillary condensation can be tested in this and similar models. After reviewing the qualitative, phenomenological description of these phenomena on a mean field level, we will summarize predictions of scaling theories. Comments will be made about the models studied and simulation techniques as well as the specific problems that occur in the relevant finite size scaling analysis. The resulting simulational data have prompted considerable new theoretical efforts, but there are still unsolved problems with respect to critical wetting. We will also present results for interface localization-delocalization transitions in both Ising models and lattice polymer mixtures in a thin film geometry and show that theory can account for many, but not all, aspects of the simulations. In systems with asymmetric boundary fields rather complex phase diagrams can result, and these should be relevant for corresponding experiments. The simulational evidence is fully compatible with the scaling predictions of Fisher and Nakanishi [J. Chem. Phys.
75:5875 (1981)] on capillary condensation. To conclude we shall summarize the major unanswered theoretical questions in this rich field of inquiry.
145 citations
TL;DR: This work investigates several three-dimensional lattice models believed to be in the Ising universality class by means of Monte Carlo methods and finite-size scaling, and analyzes all the data simultaneously such that the universal parameters occur only once, leading to an improved accuracy.
Abstract: We investigate several three-dimensional lattice models believed to be in the Ising universality class by means of Monte Carlo methods and finite-size scaling. These models include spin- 1 models with nearestneighbor interactions on the simple-cubic and on the diamond lattice. For the simple cubic lattice, we also include models with third-neighbor interactions of varying strength, and some ‘‘equivalent-neighbor’’ models. Also included are a spin-1 model and a hard-core lattice gas. Separate analyses of the numerical data confirm the Ising-like critical behavior of these systems. On this basis, we analyze all these data simultaneously such that the universal parameters occur only once. This leads to an improved accuracy. The thermal, magnetic, and irrelevant exponents are determined as y t51.5868(3), y h52.4816(1), and y i520.821(5), respectively. The Binder ratio is estimated as Q5^m 2 & 2 /^m 4 &50.62 341(4).
132 citations
TL;DR: In this article, a phenomenological theory of phase coexistence of finite systems near the coexistence curve that occurs in the thermodynamic limit is formulated for the generic case of d-dimensional ferromagnetic Ising lattices of linear dimension L with magnetization m slightly less than mcoex.
Abstract: A phenomenological theory of phase coexistence of finite systems near the coexistence curve that occurs in the thermodynamic limit is formulated for the generic case of d-dimensional ferromagnetic Ising lattices of linear dimension L with magnetization m slightly less than mcoex. It is argued that in the limit L→∞ an unconventional first-order transition occurs at a characteristic value mt
129 citations
TL;DR: In this article, it was shown that the metastable states which are relevant for the out-of-equilibrium dynamics of such systems are always in a FRSB phase.
Abstract: The low-temperature phase of discontinuous mean-field spin glasses is generally described by a one-step replica symmetry breaking (1RSB) ansatz. The Gardner transition, i.e. a very-low-temperature phase transition to a full replica symmetry breaking (FRSB) phase, is often regarded as an inessential, and somehow exotic phenomenon. In this paper we show that the metastable states which are relevant for the out-of-equilibrium dynamics of such systems are always in a FRSB phase. The only exceptions are (to the best of our knowledge) the p-spin spherical model and the random energy model (REM). We also discuss the consequences of our results for aging dynamics and for local search algorithms in hard combinatorial problems.
127 citations
TL;DR: In this paper, the authors present a detailed theoretical overview of the thermodynamic properties of the dipolar spin ice model, which has been shown to be an excellent quantitative descriptor of the Ising pyrochlore materials Dy 2Ti_2O_7 and Ho 2Ti-2O-7 and compare their predictions with experimental results.
Abstract: We present a detailed theoretical overview of the thermodynamic properties of the dipolar spin ice model, which has been shown to be an excellent quantitative descriptor of the Ising pyrochlore materials Dy_2Ti_2O_7 and Ho_2Ti_2O_7. We show that the dipolar spin ice model can reproduce an effective quasi macroscopically degenerate ground state and spin-ice behavior of these materials when the long-range nature of dipole-dipole interaction is handled carefully using Ewald summation techniques. This degeneracy is, however, ultimately lifted at low temperature. The long-range ordered state is identified via mean field theory and Monte Carlo simulation techniques. Finally, we investigate the behavior of the dipolar spin ice model in an applied magnetic field, and compare our predictions with experimental results. We find that a number of different long-range ordered states are favored by the model depending on field direction.
TL;DR: In this article, the authors compare the predictions of the effective string description of confinement with a set of Montecarlo data for the 3D gauge Ising model at finite temperature, and find that for large enough distances and low enough temperature the data are well described by a pure bosonic string.
Abstract: We compare the predictions of the effective string description of confinement with a set of Montecarlo data for the 3d gauge Ising model at finite temperature. Thanks to a new algorithm which makes use of the dual symmetry of the model we can reach very high precisions even for large quark-antiquark distances. We are thus able to explore the large R regime of the effective string. We find that for large enough distances and low enough temperature the data are well described by a pure bosonic string. As the temperature increases higher order corrections become important and cannot be neglected even at large distances. These higher order corrections seem to be well described by the Nambu-Goto action truncated at the first perturbative order.
TL;DR: In this paper, spontaneous symmetry breaking does not modify the ground-state entanglement of two spins, as defined by the concurrence, in the XXZ chain and the transverse field Ising chain.
Abstract: It is shown that spontaneous symmetry breaking does not modify the ground-state entanglement of two spins, as defined by the concurrence, in the XXZ chain and the transverse field Ising chain. Correlation function inequalities, valid in any dimensions for these models, are presented outlining the regimes where entanglement is unaffected by spontaneous symmetry breaking.
TL;DR: In this article, the ordering process of a two-dimensional Ising triangular antiferromagnet is described, and the possibility of one-dimensional features at high T is discussed.
Abstract: Specific heat and magnetization measurements have been carried out under a range of magnetic fields on single crystals of ${\mathrm{Ca}}_{3}{\mathrm{Co}}_{2}{\mathrm{O}}_{6}.$ This compound is composed of Ising magnetic chains that are arranged on a triangular lattice. The intrachain and interchain couplings are ferromagnetic and antiferromagnetic, respectively. This situation gives rise to geometrical frustration, that bears some similarity to the classical problem of a two-dimensional Ising triangular antiferromagnet. This paper reports on the ordering process at low T and the possibility of one-dimensional features at high T.
TL;DR: In this article, universal and nonuniversal critical exponents of a three-dimensional Ising system with weak quenched disorder are discussed, with a focus on the analysis of divergent series obtained.
Abstract: Universal and nonuniversal critical exponents of a three-dimensional Ising system with weak quenched disorder are discussed. Experimental, computational, and theoretical results are reviewed. Particular attention is given to field-theoretical renormalization-group results. Different renormalization schemes are considered with emphasis on the analysis of the divergent series obtained.
TL;DR: The influence of quenched disorder on quantum phase transitions in systems with overdamped dynamics is studied and the behavior based on Lifshitz-tail arguments is discussed and illustrated by simulations of a model system.
Abstract: We study the influence of quenched disorder on quantum phase transitions in systems with over-damped dynamics. For Ising order-parameter symmetry disorder destroys the sharp phase transition by rounding because a static order parameter can develop on rare spatial regions. This leads to an exponential dependence of the order parameter on the coupling constant. At finite temperatures the static order on the rare regions is destroyed. This restores the phase transition and leads to a double-exponential relation between critical temperature and coupling strength. We discuss the behavior based on Lifshitz-tail arguments and illustrate the results by simulations of a model system.
TL;DR: In this article, the authors considered the 3D Ising model for a faceted crystal and showed that in the limit q→1, a deterministic shape is attained, which has the three facets (100), (010), (001), and a rounded piece interpolating between them.
Abstract: A statistical mechanics model for a faceted crystal is the 3D Ising model at zero temperature. It is assumed that in one octant all sites are occupied by atoms, the remaining ones being empty. Allowed atom configurations are such that they can be obtained from the filled octant through successive removals of atoms with breaking of precisely three bonds. If V denotes the number of atoms removed, then the grand canonical Boltzmann weight is q
V
, 0
TL;DR: In this article, the exact formulation for the mixed spin -1 2 and spin - 3 2 Blume-Capel Ising ferrimagnetic system on the Bethe lattice by the use of exact recursion relations is presented.
Abstract: We present the exact formulation for the mixed spin - 1 2 and spin - 3 2 Blume–Capel Ising ferrimagnetic system on the Bethe lattice by the use of exact recursion relations. The exact expressions for the magnetization, quadrupole moment, Curie temperature and free energy are found and the phase diagrams are illustrated on the Bethe lattice with the coordination numbers q = 3 , 4, 5 and 6. It is found that the phase diagram of this mixed spin system only presents second-order phase transitions. The thermal variation of the magnetization belonging to each sublattice and the net magnetization are also presented.
TL;DR: In this paper, the authors demonstrate the presence of an extended critical phase in the transverse field Ising magnet on the triangular lattice, in a regime where both thermal and quantum fluctuations are important.
Abstract: We demonstrate the presence of an extended critical phase in the transverse field Ising magnet on the triangular lattice, in a regime where both thermal and quantum fluctuations are important We
TL;DR: In this article, an exact expression of the response function to an infinitesimal magnetic field for an Ising-Glauber-like model with arbitrary exchange couplings is derived in terms of thermodynamic averages.
Abstract: We derive an exact expression of the response function to an infinitesimal magnetic field for an Ising–Glauber-like model with arbitrary exchange couplings. The result is expressed in terms of thermodynamic averages and does not depend on the initial conditions or on the dimension of the space. The response function is related to time-derivatives of a complicated correlation function and so the expression is a generalization of the equilibrium fluctuation–dissipation theorem in the special case of this model. Correspondence with the Ising–Glauber model is discussed. A discrete-time version of the relation is implemented in Monte Carlo simulations and then used to study the ageing regime of the ferromagnetic two-dimensional Ising–Glauber model quenched from the paramagnetic phase to the ferromagnetic one. Our approach has the originality to give direct access to the response function and the fluctuation–dissipation ratio.
01 Mar 2003
TL;DR: In this article, the authors identify all states which are resonantly coupled to the Mott insulator via an infinitesimal tunneling amplitude between neighboring potential wells and describe the strong correlation between these states is described by an effective Hamiltonian for the resonant subspace.
Abstract: Recent experiments on ultracold atomic gases in an optical lattice potential have produced a Mott insulating state of ${}^{87}\mathrm{Rb}$ atoms. This state is stable to a small applied potential gradient (an ``electric'' field), but a resonant response was observed when the potential energy drop per lattice spacing $(E),$ was close to the repulsive interaction energy (U) between two atoms in the same lattice potential well. We identify all states which are resonantly coupled to the Mott insulator for $E\ensuremath{\approx}U$ via an infinitesimal tunneling amplitude between neighboring potential wells. The strong correlation between these states is described by an effective Hamiltonian for the resonant subspace. This Hamiltonian exhibits quantum phase transitions associated with an Ising density wave order and with the appearance of superfluidity in the directions transverse to the electric field. We suggest that the observed resonant response is related to these transitions and propose experiments to directly detect the order parameters. The generalizations to electric fields applied in different directions and to a variety of lattices should allow study of numerous other correlated quantum phases.
TL;DR: A quantum Monte Carlo algorithm for the transverse Ising model with arbitrary short- or long-range interactions is presented, based on sampling the diagonal matrix elements of the power-series expansion of the density matrix (stochastic series expansion), and avoids the interaction summations necessary in conventional methods.
Abstract: A quantum Monte Carlo algorithm for the transverse Ising model with arbitrary short- or long-range interactions is presented. The algorithm is based on sampling the diagonal matrix elements of the power-series expansion of the density matrix (stochastic series expansion), and avoids the interaction summations necessary in conventional methods. In the case of long-range interactions, the scaling of the computation time with the system size N is therefore reduced from ${N}^{2}$ to $N\mathrm{ln}(N).$ The method is tested on a one-dimensional ferromagnet in a transverse field, with interactions decaying as ${1/r}^{2}.$
TL;DR: In this paper, the antiferromagnetic Ising chain in a transverse magnetic field and uniform longitudinal field is determined using the density-matrix renormalization group calculation combined with a finite-size scaling the ground-state phase diagram in the plane.
Abstract: We have studied the antiferromagnetic Ising chain in a transverse magnetic field ${h}_{x}$ and uniform longitudinal field ${h}_{z}.$ Using the density-matrix renormalization group calculation combined with a finite-size scaling the ground-state phase diagram in ${(h}_{x}{,h}_{z})$ plane is determined. It is shown that there is an order-disordered transition line in this plane and the critical properties belong to the universality class of the two-dimensional Ising model. Based on the perturbation theory in ${h}_{z}$ the scaling behavior of the mass gap in the vicinity of the critical point ${(h}_{x}{=1/2,h}_{z}=0)$ is established. It is found that the form of the transition line near the classical multicritical point ${(h}_{x}{=0,h}_{z}=1)$ is linear. The connection of the considered quantum model with the quasi-one-dimensional classical Ising model in the magnetic field is discussed.
TL;DR: Mean-field kinetic equations as mentioned in this paper are a valuable tool to study the atomic dynamics and spin dynamics of simple lattice gas and Ising models and can be derived from the microscopic master equation of the system and contain analytical expressions for kinetic coefficients and thermodynamic quantities.
Abstract: Mean-field kinetic equations are a valuable tool to study the atomic dynamics and spin dynamics of simple lattice gas and Ising models. They can be derived from the microscopic master equation of the system and contain analytical expressions for kinetic coefficients and thermodynamic quantities which are usually introduced phenomenologically. We review several methods to obtain such equations, and discuss applications to the dynamics of order–disorder transitions, spinodal decomposition, and dendritic growth in the isothermal or chemical model. In the case of dendritic growth we show that the mean-field kinetic equations are equivalent to standard continuum equations for this problem and derive expressions for macroscopic quantities, e.g. the surface tension and kinetic coefficients, as functions of the microscopic order parameters. In spinodal decomposition, we focus our attention on the vacancy mechanism, which is a more faithful picture of diffusion in solids than the more widely examined exchange mech...
TL;DR: In this article, a new definition of the spontaneous magnetization was proposed for the two-dimensional Ising model with free boundary conditions, and the model was shown to be stable.
Abstract: We suggest a new definition of the spontaneous magnetization. We calculate this spontaneous magnetization for the two-dimensional Ising model with free boundary conditions.
TL;DR: In this article, an approximate mapping to a fully frustrated transverse-field Ising model on the dual dice lattice was obtained by exploiting the low-energy singlet excitations found below the spin gap in numerical studies of the Kagome lattice quantum Heisenberg antiferromagnet.
Abstract: This paper is concerned with physics of the low-energy singlet excitations found to exist below the spin gap in numerical studies of the Kagome lattice quantum Heisenberg antiferromagnet. Insight into the nature of these excitations is obtained by exploiting an approximate mapping to a fully frustrated transverse-field Ising model on the dual dice lattice. This Ising model is shown to possess at least two phases---an ordered phase that also breaks translational symmetry with a large unit cell, and a paramagnetic phase. The former is argued to be a likely candidate for the ground state of the original Kagome magnet which thereby exhibits a specific pattern of dimer ordering with a large unit cell. Comparisons with available numerical results are made.
TL;DR: The subject of this article is spin‐systems as studied in statistical physics, focusing on the case of two spins, and shows that there is a fully polynomial randomised approximation scheme (FPRAS) for the partition function in the “ferromagnetic” region βγ ≥ 1, but (unless RP = NP) there is no FPRAS in the antiferromagnetic region corresponding to the square.
Abstract: The subject of this article is spin-systems as studied in statistical physics. We focus on the case of two spins. This case encompasses models of physical interest, such as the classical Ising model (ferromagnetic or antiferromagnetic, with or without an applied magnetic field) and the hard-core gas model. There are three degrees of freedom, corresponding to our parameters beta, gamma and mu. We wish to study the complexity of (approximately) computing the partition function in terms of these parameters. We pay special attention to the symmetric case mu=1 for which our results are depicted in Figure 1. Exact computation of the partition function Z is NP-hard except in the trivial case beta gamma=1, so we concentrate on the issue of whether Z can be computed within small relative error in polynomial time. We show that there is a fully polynomial randomised approximation scheme (FPRAS) for the partition function in the "ferromagnetic" region beta gamma >= 1, but (unless RP=NP) there is no FPRAS in the "antiferromagnetic" region corresponding to the square defined by 0
TL;DR: In this paper, the internal dynamics of bosonic atoms in an optical lattice was studied, and it was shown that the atoms behave as localized spins which interact according to some spin Hamiltonian.
Abstract: We study the internal dynamics of bosonic atoms in an optical lattice. Within the regime in which the atomic crystal is a Mott insulator with one atom per well, the atoms behave as localized spins which interact according to some spin Hamiltonian. The type of Hamiltonian (Heisenberg, Ising), and the sign of interactions may be tuned by changing the properties of the optical lattice, or applying external magnetic fields. When, on the other hand, the number of atoms per lattice site is unknown, we can still use the bosons to perform general quantum computation.
TL;DR: In this article, the authors study the finite-temperature behavior of two-dimensional Heisenberg antiferromagnets with very weak easy-axis and easy-plane exchange anisotropies.
Abstract: We study the finite-temperature behavior of two-dimensional $S=1/2$ Heisenberg antiferromagnets with very weak easy-axis and easy-plane exchange anisotropies. By means of quantum Monte Carlo simulations, based on the continuous-time loop and worm algorithm, we obtain a rich set of data that allows us to draw conclusions about both the existence and the type of finite-temperature transition expected in the considered models. We observe that the essential features of the Ising universality class, as well as those of the Berezinskii-Kosterlitz-Thouless (BKT) one, are preserved even for anisotropies as small as ${10}^{\ensuremath{-}3}$ times the exchange integral; such outcome, being referred to the most quantum case $S=1/2,$ rules out the possibility for quantum fluctuations to destroy the long-range or quasi-long-range order, whose onset is responsible for the Ising or BKT transition, no matter how small the anisotropy. Besides this general issue, we use our results to extract, out of the isotropic component, the features which are peculiar to weakly anisotropic models, with particular attention for the temperature region immediately above the transition. By this analysis we aim to give a handy tool for understanding the experimental data relative to those real compounds whose anisotropies are too weak for a qualitative description to accomplish the goal of singling out the genuinely two-dimensional critical behavior.
TL;DR: Strong numerical evidence is provided that this transition leading to a finite-temperature phase transition is in the 2D Ising universality class, and that T(c)-->0 with an infinite slope when J2/J1-->1/2.
Abstract: We study the thermal properties of the classical antiferromagnetic Heisenberg model with both nearest (J1) and next-nearest (J2) exchange couplings on the square lattice by extensive Monte Carlo simulations. We show that, for J2/J1>1/2, thermal fluctuations give rise to an effective Z2 symmetry leading to a finite-temperature phase transition. We provide strong numerical evidence that this transition is in the 2D Ising universality class, and that T(c)-->0 with an infinite slope when J2/J1-->1/2.