Showing papers on "Ising model published in 2005"

Dynamics of a quantum phase transition: exact solution of the quantum Ising model.

Abstract: The Quantum Ising model is an exactly solvable model of quantum phase transition. This Letter gives an exact solution when the system is driven through the critical point at a finite rate. The evolution goes through a series of Landau-Zener level anticrossings when pairs of quasiparticles with opposite pseudomomenta get excited with a probability depending on the transition rate. The average density of defects excited in this way scales like a square root of the transition rate. This scaling is the same as the scaling obtained when the standard Kibble-Zurek mechanism of thermodynamic second order phase transitions is applied to the quantum phase transition in the Ising model.

Universal adiabatic dynamics in the vicinity of a quantum critical point

Abstract: We study temporal behavior of a quantum system under a slow external perturbation, which drives the system across a second-order quantum phase transition. It is shown that despite the conventional adiabaticity conditions are always violated near the critical point, the number of created excitations still goes to zero in the limit of infinitesimally slow variation of the tuning parameter. It scales with the adiabaticity parameter as a power related to the critical exponents $z$ and $\ensuremath{ u}$ characterizing the phase transition. We support general arguments by direct calculations for the Boson Hubbard and the transverse field Ising models.

High Curie Temperature and Nano-Scale Spinodal Decomposition Phase in Dilute Magnetic Semiconductors

Abstract: We show that spinoadal decomposition phase in dilute magnetic semiconductors (DMS) offers the possibility to have high Curie temperatures (TC) even if the magnetic exchange interaction is short ranged. The spinodal decomposition is simulated by applying the Monte Carlo method to the Ising model with realistic (ab initio) chemical pair interactions between magnetic impurities in DMS. Curie temperatures are estimated by the random phase approximation with taking disorder into account. It is found that the spinodal decomposition phase inherently occurs in DMS due to strong attractive interactions between impurities. This phase decomposition supports magnetic network over the dimension of the crystal resulting in a high-TC phase.

Glauber dynamics on trees and hyperbolic graphs

Abstract: We study continuous time Glauber dynamics for random configurations with local constraints (eg proper coloring, Ising and Potts models) on finite graphs with n vertices and of bounded degree We show that the relaxation time (defined as the reciprocal of the spectral gap |λ1−λ2|) for the dynamics on trees and on planar hyperbolic graphs, is polynomial in n For these hyperbolic graphs, this yields a general polynomial sampling algorithm for random configurations We then show that for general graphs, if the relaxation time τ2 satisfies τ2=O(1), then the correlation coefficient, and the mutual information, between any local function (which depends only on the configuration in a fixed window) and the boundary conditions, decays exponentially in the distance between the window and the boundary For the Ising model on a regular tree, this condition is sharp

Magnetic relaxation in rare-earth oxide pyrochlores

Abstract: A theory of magnetic relaxation is developed for geometrically frustrated three-dimensional magnets that can be described by an antiferromagnetic Ising model. These magnetic materials are exemplified by some of the recently synthesized rare-earth oxide pyrochlores, such as Dy2Ti2O7, Ho2Ti2O7, or Yb2Ti2O7. A model based on an analogy between the spin ordering in Ising magnets and proton ordering in ice is proposed. In this model, magnetic point defects treated as noninteracting quasiparticles characterized by well-defined energies, mobilities, and effective magnetic charges play a fundamental role analogous to that of ion defects in the physics of ice or by electrons and holes in semiconductors. The proposed model is used to derive expressions for magnetic susceptibility as a function of frequency and temperature.

Multiple Schramm-Loewner Evolutions and Statistical Mechanics Martingales

Abstract: A statistical mechanics argument relating partition functions to martingales is used to get a condition under which random geometric processes can describe interfaces in 2d statistical mechanics at criticality. Requiring multiple SLEs to satisfy this condition leads to some natural processes, which we study in this note. We give examples of such multiple SLEs and discuss how a choice of conformal block is related to geometric configuration of the interfaces and what is the physical meaning of mixed conformal blocks. We illustrate the general ideas on concrete computations, with applications to percolation and the Ising model

Power-law spin correlations in pyrochlore antiferromagnets

Abstract: The ground state ensemble of the highly frustrated pyrochlore-lattice antiferromagnet can be mapped to a coarse-grained ``polarization'' field satisfying a zero-divergence condition. From this it follows that the correlations of this field, as well as the actual spin correlations, decay with separation like a dipole-dipole interaction $(1∕\ensuremath{\mid}R{\ensuremath{\mid}}^{3})$. Furthermore, a lattice version of the derivation gives an approximate formula for spin correlations, with several features that agree well with simulations and neutron-diffraction measurements of diffuse scattering, in particular the pinch-point (pseudo-dipolar) singularities at reciprocal lattice vectors. This system is compared to others in which constraints also imply diffraction singularities, and other possible applications of the coarse-grained polarization are discussed.

Introduction to the Replica Theory of Disordered Statistical Systems

Abstract: This book describes the statistical mechanics of classical spin systems with quenched disorder. The first part of the book covers the physics of spin-glass states using results obtained within the framework of the mean field theory of spin glasses. The technique of replica symmetry breaking is explained in detail, along with a discussion of the underlying physics. The second part is devoted to the theory of critical phenomena in the presence of weak quenched disorder. This includes a systematic derivation of the traditional renormalization group theory, which is then used to obtain a new 'random' critical regime in disordered vector ferromagnets and in the two-dimensional Ising model. The third part of the book describes other types of disordered systems, relating to new results at the frontiers of modern research. The book is suitable for graduate students and researchers in the field of statistical mechanics of disordered systems.

Breaking of ergodicity and long relaxation times in systems with long-range interactions.

Abstract: The thermodynamic and dynamical properties of an Ising model with both short-range and long-range, mean-field-like, interactions are studied within the microcanonical ensemble. It is found that the relaxation time of thermodynamically unstable states diverges logarithmically with system size. This is in contrast with the case of short-range interactions where this time is finite. Moreover, at sufficiently low energies, gaps in the magnetization interval may develop to which no microscopic configuration corresponds. As a result, in local microcanonical dynamics the system cannot move across the gap, leading to breaking of ergodicity even in finite systems. These are general features of systems with long-range interactions and are expected to be valid even when the interaction is slowly decaying with distance.

Critical Casimir effect in three-dimensional Ising systems: measurements on binary wetting films.

Abstract: The critical Casimir force (CF) is observed in thin wetting films of a binary liquid mixture close to the liquid/vapor coexistence. X-ray reflectivity shows thickness (L) enhancement near the bulk consolute point. The extracted Casimir amplitude Delta(+-)=3+/-1 agrees with the theoretical universal value for the antisymmetric 3D Ising films. The onset of CF in the one-phase region occurs at L/xi approximately 5 regardless of whether the bulk correlation length xi is varied with temperature or composition. The shape of the Casimir scaling function depends monotonically on the dimensionality.

Comparison of voter and Glauber ordering dynamics on networks.

Abstract: We study numerically the ordering process of two very simple dynamical models for a two-state variable on several topologies with increasing levels of heterogeneity in the degree distribution. We find that the zero-temperature Glauber dynamics for the Ising model may get trapped in sets of partially ordered metastable states even for finite system size, and this becomes more probable as the size increases. Voter dynamics instead always converges to full order on finite networks, even if this does not occur via coherent growth of domains. The time needed for order to be reached diverges with the system size. In both cases the ordering process is rather insensitive to the variation of the degreee distribution from sharply peaked to scale free.

Sznajd model and its applications

Abstract: Institute of Theoretical Physics, University of Wroc law, pl. Maxa Borna 9, 50-204Wroc law, PolandIn 2000 we proposed a sociophysics model of opinion formation, whichwas based on trade union maxim ”United we Stand, Divided we Fall”(USDF) and latter due to Dietrich Stauffer became known as the Sznajdmodel (SM). The main difference between SM compared to voter or Ising-type models is that information flows outward. In this paper we reviewthe modifications and applications of SM that have been proposed in theliterature.PACS numbers: 05.50.+q Lattice theory and statistics (Ising, Potts, etc.)89.65.-s Social systems

Quantum phase transitions in the sub-Ohmic spin-boson model: failure of the quantum-classical mapping.

Abstract: The effective theories for many quantum phase transitions can be mapped onto those of classical transitions. Here we show that the naive mapping fails for the sub-Ohmic spin-boson model which describes a two-level system coupled to a bosonic bath with power-law spectral density, $J(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{s}$. Using an $ϵ$ expansion we prove that this model has a quantum transition controlled by an interacting fixed point at small $s$, and support this by numerical calculations. In contrast, the corresponding classical long-range Ising model is known to display mean-field transition behavior for $0lsl1/2$, controlled by a noninteracting fixed point. The failure of the quantum-classical mapping is argued to arise from the long-ranged interaction in imaginary time in the quantum model.

Slow relaxation in a one-dimensional rational assembly of antiferromagnetically-coupled [Mn4] single-molecule-magnets

Abstract: Four discrete MnIII/MnII tetra-nuclear complexes with double-cuboidal core were synthesized. dc magnetic measurements show that both Mn2+ - Mn3+ and Mn3+ - Mn3+ magnetic interactions are ferromagnetic in three samples leading to an S = 9 ground state for the Mn4 unit. Furthermore, these complexes are Single-Molecule Magnets (SMMs) clearly showing both thermally activated and ground state tunneling regimes. Slight changes in the [Mn4] core geometry result in an S = 1 ground state in fourth sample. A one-dimensional assembly of [Mn4] units was obtained in the same synthetic conditions with the subsequent addition of NaN3. Double chair-like N3- bridges connect identical [Mn4] units into a chain arrangement. This material behaves as an Ising assembly of S = 9 tetramers weakly antiferromagnetically coupled. Slow relaxation of the magnetization is observed at low temperature for the first time in an antiferromagnetic chain, following an activated behavior with 47 K and tau_0 = 7x10^-11 s. The observation of this original thermally activated relaxation process is induced by finite-size effects and in particular by the non-compensation of spins in segments of odd-number units. Generalizing the known theories on the dynamic properties of poly-disperse finite segments of antiferromagnetically coupled Ising spins, the theoretical expression of the characteristic energy gaps were estimated and successfully compared to the experimental values.

Chaotic properties of systems with Markov dynamics.

Abstract: We present a general approach for computing the dynamic partition function of a continuous-time Markov process. The Ruelle topological pressure is identified with the large deviation function of a physical observable. We construct for the first time a corresponding finite Kolmogorov-Sinai entropy for these processes. Then, as an example, the latter is computed for a symmetric exclusion process. We further present the first exact calculation of the topological pressure for an N-body stochastic interacting system, namely, an infinite-range Ising model endowed with spin-flip dynamics. Expressions for the KolmogorovSinai and the topological entropies follow. In statistical mechanics, bridging the microscopics to the macroscopics remains the ultimate goal, be it in or out of equilibrium. The development of the theory of dynamical systems and of their chaotic properties has led to major advances in equilibrium and nonequilibrium statistical mechanics. All those approaches make extensive use of such concepts as Lyapunov exponents, Kolmogorov-Sinai (KS) or topological entropies, topological pressure, etc., all quite mathematical in nature, and for which very few results (even nonrigorous) are available, as far as systems with many degrees of freedom are concerned. One of the central ideas in constructing a statistical physics out of equilibrium is that of Gibbs ensembles [1] in which time is seen to play the role of the volume in traditional equilibrium statistical mechanics. A central quantity called the dynamical partition function is, in general, defined as

How to compute loop corrections to the Bethe approximation

Abstract: We introduce a method for computing corrections to the Bethe approximation for spin models on arbitrary lattices. Unlike cluster variational methods, the new approach takes into account fluctuations on all length scales. The derivation of the leading correction is explained and applied to two simple examples: the ferromagnetic Ising model on d-dimensional lattices, and the spin glass on random graphs (both in their high-temperature phases). In the first case we rederive the well known Ginzburg criterion and the upper critical dimension. In the second, we compute finite-size corrections to the free energy.

Large deviation techniques applied to systems with long-range interactions

Abstract: We discuss a method to solve models with long-range interactions in the microcanonical and canonical ensemble. The method closely follows the one introduced by R.S. Ellis, Physica D 133:106 (1999), which uses large deviation techniques. We show how it can be adapted to obtain the solution of a large class of simple models, which can show ensemble inequivalence. The model Hamiltonian can have both discrete (Ising, Potts) and continuous (HMF, Free Electron Laser) state variables. This latter extension gives access to the comparison with dynamics and to the study of non-equilibrium effects. We treat both infinite range and slowly decreasing interactions and, in particular, we present the solution of the α-Ising model in one-dimension with 0 ⩽ α < 1.

Fine grained entanglement loss along renormalization group flows

Abstract: We explore entanglement loss along renormalization group trajectories as a basic quantum information property underlying their irreversibility. This analysis is carried out for the quantum Ising chain as a transverse magnetic field is changed. We consider the ground-state entanglement between a large block of spins and the rest of the chain. Entanglement loss is seen to follow from a rigid reordering, satisfying the majorization relation, of the eigenvalues of the reduced density matrix for the spin block. More generally, our results indicate that it may be possible to prove the irreversibility along renormalization group trajectories from the properties of the vacuum only, without need to study the whole Hamiltonian.

Off-equilibrium generalization of the fluctuation dissipation theorem for Ising spins and measurement of the linear response function.

Abstract: We derive for Ising spins an off-equilibrium generalization of the fluctuation dissipation theorem, which is formally identical to the one previously obtained for soft spins with Langevin dynamics [L.F. Cugliandolo, J. Kurchan, and G. Parisi, J. Phys. I 4, 1641 (1994)]. The result is quite general and holds both for dynamics with conserved and nonconserved order parameters. On the basis of this fluctuation dissipation relation, we construct an efficient numerical algorithm for the computation of the linear response function without imposing the perturbing field, which is alternative to those of Chatelain [J. Phys. A 36, 10 739 (2003)] and Ricci-Tersenghi [Phys. Rev. E 68, 065104(R) (2003)]. As applications of the new algorithm, we present very accurate data for the linear response function of the Ising chain, with conserved and nonconserved order parameter dynamics, finding that in both cases the structure is the same with a very simple physical interpretation. We also compute the integrated response function of the two-dimensional Ising model, confirming that it obeys scaling $\ensuremath{\chi}(t,{t}_{w})\ensuremath{\simeq}{t}_{w}^{\ensuremath{-}a}f(t∕{t}_{w})$, with $a=0.26\ifmmode\pm\else\textpm\fi{}0.01$, as previously found with a different method.

Competing orders in one-dimensional spin-3/2 fermionic systems.

Abstract: Novel competing orders are found in spin-3/2 cold atomic systems in one-dimensional optical traps and lattices. In particular, the quartetting phase, a four-fermion counterpart of Cooper pairing, exists in a large portion of the phase diagram. The transition between the quartetting and singlet Cooper pairing phases is controlled by an Ising symmetry breaking in one of the spin channels. The singlet Cooper pairing phase also survives in the purely repulsive interaction regime. In addition, various charge and bond ordered phases are identified at commensurate fillings in lattice systems.

History of the Lenz-Ising Model 1920–1950: From Ferromagnetic to Cooperative Phenomena

Abstract: I chart the considerable changes in the status and conception of the Lenz-Ising model from 1920 to 1950 in terms of three phases: In the early 1920s, Lenz and Ising introduced the model in the field of ferromagnetism. Based on an exact derivation, Ising concluded that it is incapable of displaying ferromagnetic behavior, a result he erroneously extended to three dimensions. In the next phase, Lenz and Ising’s contemporaries rejected the model as a representation of ferromagnetic materials because of its conflict with the new quantum mechanics. In the third phase, from the early 1930s to the early 1940s, the model was revived as a model of cooperative phenomena. I provide more detail on this history than the earlier accounts of Brush (1967) and Hoddeson, Schubert, Heims, and Baym (1992) and question some of their conclusions. Moreover, my account differs from these in its focus on the development of the model in its capacity as a model. It examines three aspects of this development: (1) the attitudes on the degree of physical realism of the Lenz-Ising model in its representation of physical phenomena; (2) the various reasons for studying and using it; and (3) the effect of the change in its theoretical basis during the transition from the old to the new quantum mechanics. A major theme of my study is that even though the Lenz-Ising model is not fully realistic, it is more useful than more realistic models because of its mathematical tractability. I argue that this point of view, important for the modern conception of the model, is novel and that its emergence, while perhaps not a consequence of its study, is coincident with the third phase of its development.

Entanglement in spin chains and lattices with long-range Ising-type interactions.

Abstract: We consider $N$ initially disentangled spins, embedded in a ring or $d$-dimensional lattice of arbitrary geometry, which interact via some long-range Ising-type interaction. We investigate relations between entanglement properties of the resulting states and the distance dependence of the interaction in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$. We provide a sufficient condition when bipartite entanglement between blocks of $L$ neighboring spins and the remaining system saturates and determine ${S}_{L}$ analytically for special configurations. We find an unbounded increase of ${S}_{L}$ as well as diverging correlation and entanglement length under certain circumstances. For arbitrarily large $N$, we can efficiently calculate all quantities associated with reduced density operators of up to ten particles.

A deformed analogue of Onsager’s symmetry in the XXZ open spin chain

Abstract: The XXZ open spin chain with general integrable boundary conditions is shown to possess a q-deformed analogue of the Onsager's algebra as fundamental non-Abelian symmetry which ensures the integrability of the model. This symmetry implies the existence of a finite set of independent mutually commuting nonlocal operators which form an Abelian subalgebra. The transfer matrix and local conserved quantities, for instance the Hamiltonian, are expressed in terms of these nonlocal operators. It follows that Onsager's original approach of the planar Ising model can be extended to the XXZ open spin chain.

Langevin description of critical phenomena with two symmetric absorbing states.

Abstract: On the basis of general considerations, we propose a Langevin equation accounting for critical phenomena occurring in the presence of two symmetric absorbing states. We study its phase diagram by mean-field arguments and direct numerical integration in physical dimensions. Our findings fully account for and clarify the intricate picture known so far from the aggregation of partial results obtained with microscopic models. We argue that the direct transition from disorder to one of two absorbing states is best described as a (generalized) voter critical point and show that it can be split into an Ising and a directed percolation transition in dimensions larger than one. DOI: 10.1103/PhysRevLett.94.230601

Phase Diagram and Critical Exponents of a Dissipative Ising Spin Chain in a Transverse Magnetic Field

Abstract: We consider a one-dimensional Ising model in a transverse magnetic field coupled to a dissipative heat bath. The phase diagram and the critical exponents are determined from extensive Monte Carlo simulations. It is shown that the character of the quantum phase transition is radically altered from the corresponding nondissipative model and the double well coupled to a dissipative heat bath with linear friction. Spatial couplings and the dissipative dynamics combine to form a new quantum criticality which is independent of dissipation strength.

Phase diagrams of a nanoparticle described by the transverse Ising model

Abstract: The phase diagrams of a ferroelectric small particle described by the transverse Ising model (TIM) are investigated by the use of two theoretical frameworks, namely the standard mean-field theory and the effective-field theory corresponding to the Zernike approximation. The particle is represented by a two-dimensional array of pseudo-spins with two types of exchange interactions (J in bulk and JS on the surface) and two types of transverse fields (Ω in the bulk and ΩS on the surface). We study the phase diagrams of this model, changing the size S of the particle and the ratios of JS/J and ΩS/Ω. We find that the critical behaviors are a little different from those found for the phase diagrams of ferroelectric thin films described by the TIM. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Monte Carlo study of hard pentagons

Abstract: How does a liquid freeze if the geometry of its particles conflicts with the symmetry of the crystal it should naturally form? We study this question in the simplest model system of particles exhibiting such a symmetry mismatch: hard pentagons in two dimensions. Using isobaric and isotensic Monte Carlo simulations we have studied the phase behavior of hard pentagons. On increasing the pressure from the homogeneous and isotropic low-density phase, the system first exhibits a rotator phase (plastic solid) with a triangular lattice structure. At higher densities it undergoes a weak first-order phase transition into a "striped" phase composed of alternating rows of oppositely pointing particles. This phase is analogous to the "striped" phase in the compressible antiferromagnetic Ising model on a triangular lattice and is an example of systems in which frustration due to the mismatch in symmetries is released by an elastic coupling to the lattice. In order to pursue this analogy we also consider hard heptagons, showing that in this case the decrease in symmetry mismatch indeed leads to a shift in the transition densities to higher values and a weakening of the transition.

Ward Identities and Chiral Anomaly in the Luttinger Liquid

Abstract: Systems of interacting non relativistic fermions in d = 1, as well as spin chains or interacting bidimensional Ising models, verify an hidden approximate Gauge invariance which can be used to derive suitable Ward identities. Despite the presence of corrections and anomalies, such Ward identities can be implemented in a Renormalization Group approach and used to exploit non- trivial cancellations which allow to control the ow of the running coupling constants; in particular this is achieved combining Ward identities, Dyson equations and suitable correction identities for the extra terms appearing in the Ward identities, due to the presence of cutos breaking the local gauge symmetry. The correlations can be computed and show a Luttinger liquid behavior charac- terized by non universal critical indices, so that the general Luttinger liquid construction for one dimensional systems is completed without any use of exact solutions. The ultraviolet cuto can be removed and a Quantum Field Theory corresponding to the Thirring model is also constructed. 1.1 Luttinger liquids. A key notion in solid state physics is the one of Fermi liquids, used to describe systems of inter- acting electrons which, in spite of the interaction, have a physical behavior qualitatively similar to the one of the free Fermi gas. In analogy with Fermi liquids, the notion of Luttinger liquids has been more recently introduced, to describe systems behaving qualitatively as the Luttinger model, see for instance (A) or (Af); their correlations have an anomalous behavior described in terms of non universal (i.e. nontrivial functions of the coupling) critical indices. A large number of models, for which an exact solution is lacking (at least for the correlation functions) are indeed believed to be in the same class of universality of the Luttinger model or of its massive version, and indeed in the last decade, starting from (BG), it has been possible to substantiate this assumption on a large class of models, by a quantitative analysis based on Renor- malization Group techniques, which at the end allow us to write the correlations and the critical indices as convergent series in the coupling. We mention the Schwinger functions of interacting non relativistic fermions in d = 1 (modelling the electronic properties of metals so anisotropic to be considered as one dimensional), in the spinless (BGPS), in the spinning case with repulsive interac- tion (BoM), or with external periodic or quasi-periodic potentials (M); the spin-spin correlations of the Heisenberg XYZ spin chain (BM1), (BM2); the thermodynamic functions of classical Ising sys- tems on a bidimensional lattice with quartic interactions like the Eight-vertex or the Ashkin-Teller models (M1); and many others, see for instance the review (GM). In all such models the observables are written as Grassmannian integrals, and a naive evaluation of them in terms of a series expansion in the perturbative parameter does not work; it is however possible, by a multiscale analysis based on Renormalization Group, to write the Grassmannian integrals as series of suitable nitely many parameters, called running coupling constants, and this expansion is convergent if the running coupling constants are small. The running coupling constants obey a complicated set of recursive equations, whose right hand side is called, as usual, the Beta function. The Beta function can be written as sum of two terms; one, which we call principal part of the Beta function, is common to all such models while the other one is model dependent. It