Showing papers on "Ising model published in 2006"

Decay of Loschmidt echo enhanced by quantum criticality.

Abstract: We study the transition of a quantum system S from a pure state to a mixed one, which is induced by the quantum criticality of the surrounding system E coupled to it. To characterize this transition quantitatively, we carefully examine the behavior of the Loschmidt echo (LE) of E modeled as an Ising model in a transverse field, which behaves as a measuring apparatus in quantum measurement. It is found that the quantum critical behavior of E strongly affects its capability of enhancing the decay of LE: near the critical value of the transverse field entailing the happening of quantum phase transition, the off-diagonal elements of the reduced density matrix describing S vanish sharply.

The Random-Cluster Model

Abstract: The class of random-cluster models is a unification of a variety of stochastic processes of significance for probability and statistical physics, including percolation, Ising, and Potts models; in addition, their study has impact on the theory of certain random combinatorial structures, and of electrical networks. Much (but not all) of the physical theory of Ising/Potts models is best implemented in the context of the random-cluster representation. This systematic summary of random-cluster models includes accounts of the fundamental methods and inequalities, the uniqueness and specification of infinite-volume measures, the existence and nature of the phase transition, and the structure of the subcritical and supercritical phases. The theory for two-dimensional lattices is better developed than for three and more dimensions. There is a rich collection of open problems, including some of substantial significance for the general area of disordered systems, and these are highlighted when encountered. Amongst the major open questions, there is the problem of ascertaining the exact nature of the phase transition for general values of the cluster-weighting factor q, and the problem of proving that the critical random-cluster model in two dimensions, with 1 ≤ q ≤ 4, converges when re-scaled to a stochastic Lowner evolution (SLE). Overall the emphasis is upon the random-cluster model for its own sake, rather than upon its applications to Ising and Potts systems.

Obtaining reaction coordinates by likelihood maximization.

Abstract: We present a new approach for calculating reaction coordinates in complex systems. The new method is based on transition path sampling and likelihood maximization. It requires fewer trajectories than a single iteration of existing procedures, and it applies to both low and high friction dynamics. The new method screens a set of candidate collective variables for a good reaction coordinate that depends on a few relevant variables. The Bayesian information criterion determines whether additional variables significantly improve the reaction coordinate. Additionally, we present an advantageous transition path sampling algorithm and an algorithm to generate the most likely transition path in the space of collective variables. The method is demonstrated on two systems: a bistable model potential energy surface and nucleation in the Ising model. For the Ising model of nucleation, we quantify for the first time the role of nuclei surface area in the nucleation reaction coordinate. Surprisingly, increased surface area increases the stability of nuclei in two dimensions but decreases nuclei stability in three dimensions.

Rare region effects at classical, quantum, and non-equilibrium phase transitions

Abstract: Rare regions, i.e., rare large spatial disorder fluctuations, can dramatically change the properties of a phase transition in a quenched disordered system. In generic classical equilibrium systems, they lead to an essential singularity, the so-called Griffiths singularity, of the free energy in the vicinity of the phase transition. Stronger effects can be observed at zero-temperature quantum phase transitions, at nonequilibrium phase transitions, and in systems with correlated disorder. In some cases, rare regions can actually completely destroy the sharp phase transition by smearing. This topical review presents a unifying framework for rare region effects at weakly disordered classical, quantum, and nonequilibrium phase transitions based on the effective dimensionality of the rare regions. Explicit examples include disordered classical Ising and Heisenberg models, insulating and metallic random quantum magnets, and the disordered contact process.

Rare region effects at classical, quantum and nonequilibrium phase transitions

Abstract: Rare regions, i.e., rare large spatial disorder fluctuations, can dramatically change the properties of a phase transition in a quenched disordered system. In generic classical equilibrium systems, they lead to an essential singularity, the so-called Griffiths singularity, of the free energy in the vicinity of the phase transition. Stronger effects can be observed at zero-temperature quantum phase transitions, at nonequilibrium phase transitions and in systems with correlated disorder. In some cases, rare regions can actually completely destroy the sharp phase transition by smearing. This topical review presents a unifying framework for rare region effects at weakly disordered classical, quantum and nonequilibrium phase transitions based on the effective dimensionality of the rare regions. Explicit examples include disordered classical Ising and Heisenberg models, insulating and metallic random quantum magnets, and the disordered contact process.

Towards conformal invariance of 2D lattice models

Abstract: Many 2D lattice models of physical phenomena are conjectured to have conformally invariant scaling limits: percolation, Ising model, self-avoiding polymers, etc. This has led to numerous exact (but non-rigorous) predictions of their scaling exponents and dimensions. We will discuss how to prove the conformal invariance conjectures, especially in relation to Schramm�Loewner evolution.

Non-equilibrium phase transitions

Abstract: These lecture notes give a basic introduction to the physics of phase transitions under non-equilibrium conditions. The notes start with a general introduction to non-equilibrium statistical mechanics followed by four parts. The first one discusses the universality class of directed percolation, which plays a similar role as the Ising model in equilibrium statistical physics. The second one gives an overview about other universality classes which have been of interest in recent years. The third part extends the scope to models with long-range interactions, including memory effects and the so-called Levy flights. Finally, the fourth part is concerned with deposition–evaporation phenomena leading to wetting transitions out of equilibrium.

Thermodynamic formalism for systems with Markov dynamics

Abstract: The thermodynamic formalism allows one to access the chaotic properties of equilibrium and out-of-equilibrium systems, by deriving those from a dynamical partition function. The definition that has been given for this partition function within the framework of discrete time Markov chains was not suitable for continuous time Markov dynamics. Here we propose another interpretation of the definition that allows us to apply the thermodynamic formalism to continuous time. We also generalize the formalism --a dynamical Gibbs ensemble construction-- to a whole family of observables and their associated large deviation functions. This allows us to make the connection between the thermodynamic formalism and the observable involved in the much-studied fluctuation theorem. We illustrate our approach on various physical systems: random walks, exclusion processes, an Ising model and the contact process. In the latter cases, we identify a signature of the occurrence of dynamical phase transitions. We show that this signature can already be unravelled using the simplest dynamical ensemble one could define, based on the number of configuration changes a system has undergone over an asymptotically large time window.

The relaxor enigma — charge disorder and random fields in ferroelectrics

Abstract: Substitutional charge disorder giving rise to quenched electric random-fields (RFs) is probably at the origin of the peculiar behavior of relaxor ferroelectrics, which are primarily characterized by their strong frequency dispersion of the dielectric response and by an apparent lack of macroscopic symmetry breaking at the phase transition. Spatial fluctuations of the RFs correlate the dipolar fluctuations and give rise to polar nanoregions in the paraelectric regime as has been evidenced by piezoresponse force microscopy (PFM) at the nanoscale. The dimension of the order parameter decides upon whether the ferroelectric phase transition is destroyed (e.g. in cubic PbMg1/3Nb2/3O3, PMN) or modified towards RF Ising model behavior (e.g. in tetragonal Sr1−x BaxNb2O6, SBN, x ≈ 0.4). Frustrated interaction between the polar nanoregions in cubic relaxors gives rise to cluster glass states as evidenced by strong pressure dependence, typical dipolar slowing-down and theoretically treated within a spherical random bond-RF model. On the other hand, freezing into a domain state takes place in uniaxial relaxors. While at T c non-classical critical behavior with critical exponents ρ ≈ 1.8, β ≈ 0.1 and α ≈ 0 is encountered in accordance with the RF Ising model, below T c ≈ 350 K RF pinning of the walls of frozen-in nanodomains gives rise to non-Debye dielectric response. It is relaxation- and creep-like at radio and very low frequencies, respectively.

Universality in three-dimensional Ising spin glasses: A Monte Carlo study

Abstract: We study universality in three-dimensional Ising spin glasses by large-scale Monte Carlo simulations of the Edwards-Anderson Ising spin glass for several choices of bond distributions, with particular emphasis on Gaussian and bimodal interactions. A finite-size scaling analysis suggests that three-dimensional Ising spin glasses obey universality.

The Theory of Magnetism Made Simple:An Introduction To Physical Concepts And To Some Useful Mathematical Methods

Abstract: The original edition of "The Theory of Magnetism" developed the various relevant topics using modern methods adapted for the many-body problem It presented and taught the fermionic field theory central to Onsager's analysis of the statistical mechanics of the two-dimensional Ising model of magnetism In its pages the Lieb-Mattis theorems on magnetic ordering of electronic energy levels and on the absence of ferromagnetism in one dimension were restated and proved in a form accessible to students The exchange mechanism in insulators and the Ruderman-Kittel interaction in metals were some of the innovative topics presented to the reader Spin waves and their interactions were analyzed in some detail In this new edition, while retaining much of the material in earlier editions, especially the first chapter, the author has eliminated some of the bulk and added a number of new subjects Among these are the effects of lowering the dimensionality (exact solutions of some important models in zero and one dimension are exhibited and contrasted with the three-dimensional versions) and the importance of the two-body Coulomb interactions The reader is introduced to the topic of critical exponents Quoting a novel theorem by Lieb and exotic band structures, the author re-examines the origins of ferromagnetism In the presentation, physical principles come first, the mathematics second Developing the reader's intuition and mastery of the subject takes precedence

Ising models for networks of real neurons

Abstract: Ising models with pairwise interactions are the least structured, or maximum-entropy, probability distributions that exactly reproduce measured pairwise correlations between spins. Here we use this equivalence to construct Ising models that describe the correlated spiking activity of populations of 40 neurons in the retina, and show that pairwise interactions account for observed higher-order correlations. By first finding a representative ensemble for observed networks we can create synthetic networks of 120 neurons, and find that with increasing size the networks operate closer to a critical point and start exhibiting collective behaviors reminiscent of spin glasses.

Series Expansion Methods for Strongly Interacting Lattice Models

Abstract: Preface 1. Introduction 2. High- and low-temperature expansions for the Ising Model 3. Models with continuous symmetry and the free graph expansion 4. Quantum spin models at T = 0 5. Quantum antiferromagnets at T = 0 6. Correlators, dynamical structure factors and multi-particle excitations 7. Quantum spin models at finite temperature 8. Electronic models 9. Review of lattice gauge theory 10. Series expansions for lattice gauge models 11. Additional topics Appendices Bibliography Index.

Adiabatic-impulse approximation for avoided level crossings: From phase-transition dynamics to Landau-Zener evolutions and back again

Abstract: We show that a simple approximation based on concepts underlying the Kibble-Zurek theory of second order phase-transition dynamics can be used to treat avoided level crossing problems. The approach discussed in this paper provides an intuitive insight into quantum dynamics of two-level systems, and may serve as a link between the theory of dynamics of classical and quantum phase transitions. To illustrate these ideas we analyze dynamics of a paramagnet-ferromagnet quantum phase transition in the Ising model. We also present exact unpublished solutions of the Landau-Zener-like problems.

Statistical-temperature Monte Carlo and molecular dynamics algorithms.

Abstract: A simulation method is presented that achieves a flat energy distribution by updating the statistical temperature instead of the density of states in Wang-Landau sampling. A novel molecular dynamics algorithm (STMD) applicable to complex systems and a Monte Carlo algorithm are developed from this point of view. Accelerated convergence for large energy bins, essential for large systems, is demonstrated in tests on the Ising model, the Lennard-Jones fluid, and bead models of proteins. STMD shows a superior ability to find local minima in proteins and new global minima are found for the 55 bead AB model in two and three dimensions. Calculations of the occupation probabilities of individual protein inherent structures provide new insights into folding and misfolding.

Inverted Berezinskii-Kosterlitz-Thouless singularity and high-temperature algebraic order in an Ising model on a scale-free hierarchical-lattice small-world network.

Abstract: We have obtained exact results for the Ising model on a hierarchical lattice incorporating three key features characterizing many real-world networks---a scale-free degree distribution, a high clustering coefficient, and the small-world effect. By varying the probability $p$ of long-range bonds, the entire spectrum from an unclustered, non-small-world network to a highly clustered, small-world system is studied. Using the self-similar structure of the network, we obtain analytic expressions for the degree distribution $P(k)$ and clustering coefficient $C$ for all $p$, as well as the average path length $\ensuremath{\ell}$ for $p=0$ and $1$. The ferromagnetic Ising model on this network is studied through an exact renormalization-group transformation of the quenched bond probability distribution, using up to $562\phantom{\rule{0.2em}{0ex}}500$ renormalized probability bins to represent the distribution. For $pl0.494$, we find power-law critical behavior of the magnetization and susceptibility, with critical exponents continuously varying with $p$, and exponential decay of correlations away from ${T}_{c}$. For $p\ensuremath{\ge}0.494$, in fact where the network exhibits small-world character, the critical behavior radically changes: We find a highly unusual phase transition, namely an inverted Berezinskii-Kosterlitz-Thouless singularity, between a low-temperature phase with nonzero magnetization and finite correlation length and a high-temperature phase with zero magnetization and infinite correlation length, with power-law decay of correlations throughout the phase. Approaching ${T}_{c}$ from below, the magnetization and the susceptibility, respectively, exhibit the singularities of $\mathrm{exp}(\ensuremath{-}C∕\sqrt{{T}_{c}\ensuremath{-}T})$ and $\mathrm{exp}(D∕\sqrt{{T}_{c}\ensuremath{-}T})$, with $C$ and $D$ positive constants. With long-range bond strengths decaying with distance, we see a phase transition with power-law critical singularities for all $p$, and evaluate an unusually narrow critical region and important corrections to power-law behavior that depend on the exponent characterizing the decay of long-range interactions.

Steplike magnetization in a spin-chain system:Ca3Co2O6.

Abstract: Because of a ferromagnetic in-chain coupling between Co3+ ions at trigonal sites, Co2O6 chains are considered as large rigid spin moments. The antiferromagnetic Ising model on the triangular lattice is applied to describe an interchain ordering. An evolution of metastable states in a sweeping magnetic field is investigated by the single-flip technique. At the first approximation two steps in the magnetization curve and a plateau at 1/3 of the saturation magnetization are found. Four steps in magnetization are determined in high-order approximations in agreement with experimental results.

On the Construction of Quantum Field Theories with Factorizing S-Matrices

Abstract: The subject of this thesis is a novel construction method for interacting relativistic quantum field theories on two-dimensional Minkowski space. The input in this construction is not a classical Lagrangian, but rather a prescribed factorizing S-matrix, i.e. the inverse scattering problem for such quantum field theories is studied. For a large class of factorizing S-matrices, certain associated quantum fields, which are localized in wedge-shaped regions of Minkowski space, are constructed explicitely. With the help of these fields, the local observable content of the corresponding model is defined and analyzed by employing methods from the algebraic framework of quantum field theory. The abstract problem in this analysis amounts to the question under which conditions an algebra of wedge-localized observables can be used to generate a net of local observable algebras with the right physical properties. The answer given here uses the so-called modular nuclearity condition, which is shown to imply the existence of local observables and the Reeh-Schlieder property. In the analysis of the concrete models, this condition is proven for a large family of S-matrices, including the scattering operators of the Sinh-Gordon model and the scaling Ising model as special examples. The so constructed models are then investigated with respect to their scattering properties. They are shown to solve the inverse scattering problem for the considered S-matrices, and a proof of asymptotic completeness is given.

Geometric frustration in the class of exactly solvable Ising–Heisenberg diamond chains

Abstract: Ground-state and finite-temperature properties of the mixed spin- and spin-S Ising–Heisenberg diamond chains are examined within an exact analytical approach based on the generalized decoration–iteration map. A particular emphasis is laid on the investigation of the effect of geometric frustration, which is generated by the competition between Heisenberg- and Ising-type exchange interactions. It is found that an interplay between the geometric frustration and quantum effects gives rise to several quantum ground states with entangled spin states in addition to some semi-classically ordered ones. Among the most interesting results to emerge from our study one could mention rigorous evidence for quantized plateaux in magnetization curves, an appearance of the round minimum in the thermal dependence of susceptibility times temperature data, double-peak zero-field specific heat curves, or an enhanced magnetocaloric effect when the frustration comes into play. The triple-peak specific heat curve is also detected when applying small external field to the system driven by the frustration into the disordered state.

Exact results of a mixed spin-1/2 and spin-S Ising model on a bathroom tile (4–8) lattice: Effect of uniaxial single-ion anisotropy

Abstract: Effect of uniaxial single-ion anisotropy upon magnetic properties of a mixed spin-1/2 and spin-S (S⩾1S⩾1) Ising model on a bathroom tile (4–8) lattice is examined within the framework of an exact star-triangle mapping transformation. Particular attention is focused on the phase diagrams established for several values of the quantum spin number S. It is shown that the mixed-spin bathroom tile lattice exhibits very similar phase boundaries as the mixed-spin honeycomb lattice whose critical points are merely slightly enhanced with respect to the former ones. The influence of uniaxial single-ion anisotropy upon the total magnetization vs. temperature dependence is particularly investigated as well.

A simple condition implying rapid mixing of single-site dynamics on spin systems

Abstract: Spin systems are a general way to describe local interactions between nodes in a graph. In statistical mechanics, spin systems are often used as a model for physical systems. In computer science, they comprise an important class of families of combinatorial objects, for which approximate counting and sampling algorithms remain an elusive goal. The Dobrushin condition states that every row sum of the "influence matrix" for a spin system is less than 1 - epsiv, where epsiv > 0. This criterion implies rapid convergence (O(n log n) mixing time) of the single-site (Glauber) dynamics for a spin system, as well as uniqueness of the Gibbs measure. The dual criterion that every column sum of the influence matrix is less than 1 - epsiv has also been shown to imply the same conclusions. We examine a common generalization of these conditions, namely that the maximum eigenvalue of the influence matrix is less than 1 epsiv. Our main result is that this criterion implies O(n log n) mixing time for the Glauber dynamics. As applications, we consider the Ising model, hard-core lattice gas model, and graph colorings, relating the mixing time of the Glauber dynamics to the maximum eigenvalue for the adjacency matrix of the graph. For the special case of planar graphs, this leads to improved bounds on mixing time with quite simple proofs

Critical point estimation of the Lennard-Jones pure fluid and binary mixtures

Abstract: The apparent critical point of the pure fluid and binary mixtures interacting with the Lennard-Jones potential has been calculated using Monte Carlo histogram reweighting techniques combined with either a fourth order cumulant calculation (Binder parameter) or a mixed-field study. By extrapolating these finite system size results through a finite size scaling analysis we estimate the infinite system size critical point. Excellent agreement is found between all methodologies as well as previous works, both for the pure fluid and the binary mixture studied. The combination of the proposed cumulant method with the use of finite size scaling is found to present advantages with respect to the mixed-field analysis since no matching to the Ising universal distribution is required while maintaining the same statistical efficiency. In addition, the accurate estimation of the finite critical point becomes straightforward while the scaling of density and composition is also possible and allows for the estimation of the line of critical points for a Lennard-Jones mixture.

Infinite-range Ising ferromagnet in a time-dependent transverse magnetic field: Quench and ac dynamics near the quantum critical point

Abstract: We study an infinite-range ferromagnetic Ising model in the presence of a transverse magnetic field, which exhibits a quantum paramagnetic-ferromagnetic phase transition at a critical value of the transverse field. In the thermodynamic limit, the low-temperature properties of this model are dominated by the behavior of a single large classical spin governed by an anisotropic Hamiltonian. Using this property, we study the quench and ac dynamics of the model both numerically and analytically, and develop a correspondence between the classical phase-space dynamics of a single spin and the quantum dynamics of the infinite-range ferromagnetic Ising model. In particular, we compare the behavior of the equal-time order-parameter correlation function both near to and away from the quantum critical point in the presence of a quench or ac transverse field. We explicitly demonstrate that a clear signature of the quantum critical point can be obtained by studying the ac dynamics of the system even in the classical limit. We discuss possible realizations of our model in experimental systems.

Dynamics of a quantum phase transition in the random Ising model : Logarithmic dependence of the defect density on the transition rate

Abstract: A quantum phase transition from paramagnetic to ferromagnetic phase is driven by a time-dependent external magnetic field. For any rate of the transition the evolution is nonadiabatic and finite density of defects is excited in the ferromagnetic state. The density of excitations has only logarithmic dependence on the transition rate. This is qualitatively different than any usual power law scaling predicted for pure systems by the Kibble-Zurek mechanism. No matter how slow the transition is the defect density remains more or less the same.

Static and dynamic critical behavior of a symmetrical binary fluid: a computer simulation.

Abstract: A symmetrical binary, A+B Lennard-Jones mixture is studied by a combination of semi-grand-canonical Monte Carlo (SGMC) and molecular dynamics (MD) methods near a liquid-liquid critical temperature Tc. Choosing equal chemical potentials for the two species, the SGMC switches identities (A→B→A) to generate well-equilibrated configurations of the system on the coexistence curve for T Tc. A finite-size scaling analysis of the concentration susceptibility above Tc and of the order parameter below Tc is performed, varying the number of particles from N=400 to 12 800. The data are fully compatible with the expected critical exponents of the three-dimensional Ising universality class. The equilibrium configurations from the SGMC runs are used as initial states for microcanonical MD runs, from which transport coefficients are extracted. Self-diffusion coefficients are obtained from the Einstein relation, while the interdiffusion coefficient and the shear viscosit...

Holomorphic parafermions in the Potts model and stochastic Loewner evolution

Abstract: We analyse parafermionic operators in the Q-state Potts model from three different perspectives First, we explicitly construct lattice holomorphic observables in the Fortuin–Kasteleyn representation, and point out some special simplifying features of the particular case Q = 2 (Ising model) In particular, away from criticality, we find a lattice generalization of the massive Majorana fermion equation We also compare the parafermionic scaling dimensions with known results from conformal field theory and Coulomb gas methods in the continuum Finally, we show that expectation values of these parafermions correspond to local observables of the stochastic Loewner evolution process which is conjectured to describe the scaling limit of the Q-state Potts model

High precision Monte Carlo simulations of interfaces in the three-dimensional Ising model: a comparison with the Nambu-Goto effective string model

Abstract: Motivated by the recent progress in the effective string description of the interquark potential in lattice gauge theory, we study interfaces with periodic boundary conditions in the three-dimensional Ising model. Our Monte Carlo results for the associated free energy are compared with the next-to-leading order (NLO) approximation of the Nambu-Goto string model. We find clear evidence for the validity of the effective string model at the level of the NLO truncation.