Thermodynamics of some layered Ising models

Thermodynamic properties of some inhomogeneous Ising models with layered distribution of couplings are studied. In particular, the specific heat is investigated in detail, both analytically and numerically. It is shown that different ordering mechanisms, namely ordering of finite regions and global ordering of infinite range, can occur in different temperature ranges. This results in remarkable structures of the specific heat curves. In addition we investigate the case where the random distribution of couplings extends over an infinite distance in one space direction. The ordinary Ising singularity then changes to one of infinite order provided the transition temperature remains finite.     

Monte Carlo studies of finite-size effects at first-order transitions

First-order phase transitions are ubiquitous in nature but their presence is often uncertain because of the effects which finite size has on all transitions. In this article we consider a general treatment of size effects on lattice systems with discrete degrees of freedom and which undergo a first-order transition in the thermodynamic limit. We review recent work involving studies of the distribution functions of the magnetization and energy at a first-order transition in a finite sample of size N connected to a bath of size N′. Two cases: N′ = ∞ and N′ = finite are considered. In the former (canonical ensemble) case, the distributions are approximated by a superposition of Gaussians corresponding to the different phases; all finite-size effects then vary as N or 1/N. The latter case involves the Gaussian ensemble where the entropy of the bath has a convenient form; for small N′, first-order transitions are characterized by van der Waals' loops in (for example) the energy vs. temperature curves. Results from extensive Monte Carlo simulations of Ising and Potts models in d = 2 are presented to confirm the predictions. Comparison is made with data from second-order transitions to show that the order of a transition can be distinguished through such studies, although problems still occur for first-order transitions very close to critical points.     

Thermodynamics of some layered Ising models

Thermodynamic properties of some inhomogeneous Ising models with layered distribution of couplings are studied. In particular, the specific heat is investigated in detail, both analytically and numerically. It is shown that different ordering mechanisms, namely ordering of finite regions and global ordering of infinite range, can occur in different temperature ranges. This results in remarkable structures of the specific heat curves. In addition we investigate the case where the random distribution of couplings extends over an infinite distance in one space direction. The ordinary Ising singularity then changes to one of infinite order provided the transition temperature remains finite.Work performed within the research program of the Sonderforschungsbereich 125 Aachen-Jülich-Köln     

Low-temperature broken-symmetry phases of spiral antiferromagnets

We study Heisenberg antiferromagnets with nearest- (J1) and third- (J3) neighbor exchange on the square lattice. In the limit of spin S-->infinity, there is a zero temperature (T) Lifshitz point at J(3)=1/4J(1), with long-range spiral spin order at T=0 for J3>1/4J(1). We present classical Monte Carlo simulations and a theory for T>0 crossovers near the Lifshitz point: spin rotation symmetry is restored at any T>0, but there is a broken lattice reflection symmetry for 0< or =T相似文献   

The Ising model on random lattices in arbitrary dimensions

We study analytically the Ising model coupled to random lattices in dimension three and higher. The family of random lattices we use is generated by the large N limit of a colored tensor model generalizing the two-matrix model for Ising spins on random surfaces. We show that, in the continuum limit, the spin system does not exhibit a phase transition at finite temperature, in agreement with numerical investigations. Furthermore we outline a general method to study critical behavior in colored tensor models.     

Rigorous analysis of singularities and absence of analytic continuation at first-order phase-transition points in lattice-spin models

We report about two new rigorous results on the nonanalytic properties of thermodynamic potentials at first-order phase transition. For lattice models (d>or=2) with arbitrary finite state space, finite-range interactions which have two ground states, we prove that the pressure has no analytic continuation at the first-order phase-transition point, under the only further assumptions that the Peierls condition is satisfied for the ground states and that the temperature is sufficiently low. For Ising models with Kac potentials J(gamma)(x)=gamma(d)phi(gammax), where 00) and analyticity in the mean field limit (gamma SE pointing arrow 0).     

Effective scaling behavior of magnetization and Hamming distance in Ising thin films under a surface field

The recent improvements on the technology for developing high-quality thin magnetic films has renewed the interest in the study of surface effects in both static and dynamic magnetic responses. In this work, we use a Monte-Carlo algorithm with Metropolis dynamics together with a spreading of damage technique to study the interplay between the effects of finite thickness and surface ordering field in thin ferromagnetic Ising (S=1/2) films. We calculate, near the bulk critical temperature and several values of the surface field, the dependence on the film thickness of the average magnetization M and Hamming distance D. We employ a finite size scaling analysis to show that both obey an effective one-parameter scaling but exhibit distinct characteristic surface fields. At their corresponding characteristic surface fields both M and D become roughly thickness independent and we estimate the critical exponent characterizing the behavior of the typical scaling lengths. Received 29 March 1999 and Received in final form 21 April 1999     

Finite-size scaling properties of the damage distance and dynamical critical exponent for the Ising model

With the damage spreading method, scaling properties of the damage distance on the Ising model with heat bath dynamics are studied numerically. With the parallel flipping scheme, the scaling curves fall on two curves, which depend on the odd or even lattice sizes. The both scaling curves give the consistent dynamical exponent as z = 2.16±0.04 for d = 2 and z = 2.09±0.05 for d = 3, respectively. By shifting one of them, two curves overlap each other perfectly. Meanwhile, all the scaling curves obtained by single-spin flipping processes (with different odd or even lattice sizes) fall on a single curve, from which the consistent dynamical critical exponent with the parallel scheme is obtained z = 2.18±0.02 for d = 2 and z = 2.08±0.04 for d = 3.     

On the damage spreading in Ising spin glasses

The spreading of the Hamming distance or damage has been investigated for ±J Ising spin glasses under heat bath dynamics. Dimensions d = 2, 3, 4, 6 and mean field were studied. For finite dimensions, the damage goes to zero at long times above a temperature T_D(d). Accurate values of this critical temperature were obtained, together with certain critical exponents. The spin glass ordering temperatures T_g were also estimated from the damage spreading data. The results are compared with other work and discussed from a phase space approach.     

Asymptotic behavior of fluctuations for the 1D Ising model in zero-temperature limit

Fluctuation of the average spin for one-dimensional Ising spins with nearest neighbor interactions are studied. The distribution function for the average spin is calculated for a finite volume, finite temperature, and finite magnetic field. As the volume increases and the temperature diminishes at zero magnetic field, there are two limits in which the probability distribution shows quite different behaviors: in the thermodynamic limit as the volume goes to infinity for finite temperature, small deviations of the fluctuations are described by a Gaussian distribution, and in the limit as the temperature vanishes for a finite volume, the ground states are realized with probability one. The crossover between these limits is analyzed via a ratio of the correlation length to the volume. The helix-coil transition in a polypeptide is discussed as an application.     

Damage spreading on networks: Clustering effects

The damage spreading of the Ising model on three kinds of networks is studied with Glauber dynamics. One of the networks is generated by evolving the hexagonal lattice with the star-triangle transformation. Another kind of network is constructed by connecting the midpoints of the edges of the topological hexagonal lattice. With the evolution of these structures, damage spreading transition temperature increases and a general explanation for this phenomenon is presented from the view of the network. The relationship between the transition temperature and the network measure-clustering coefficient is set up and it is shown that the increase of damage spreading transition temperature is the result of more and more clustering of the network. We construct the third kind of network-random graphs with Poisson degree distributions by changing the average degree of the network. We show that the increase in the average degree is equivalent to the clustering of nodes and this leads to the increase in damage spreading transition temperature.      

Planelike Interfaces in Long-Range Ising Models and Connections with Nonlocal Minimal Surfaces

This paper contains three types of results:

• the construction of ground state solutions for a long-range Ising model whose interfaces stay at a bounded distance from any given hyperplane,
• the construction of nonlocal minimal surfaces which stay at a bounded distance from any given hyperplane,
• the reciprocal approximation of ground states for long-range Ising models and nonlocal minimal surfaces.

In particular, we establish the existence of ground state solutions for long-range Ising models with planelike interfaces, which possess scale invariant properties with respect to the periodicity size of the environment. The range of interaction of the Hamiltonian is not necessarily assumed to be finite and also polynomial tails are taken into account (i.e. particles can interact even if they are very far apart the one from the other). In addition, we provide a rigorous bridge between the theory of long-range Ising models and that of nonlocal minimal surfaces, via some precise limit result.

     

Coexistence of ferromagnetism and paramagnetism in a ferromagnetic monolayer

The ferromagnetic-paramagnetic phase transition in an Fe monolayer grown on a 2 ML Au/W(110) substrate and capped with a 1 ML Au layer was studied with spin polarized low electron energy microscopy. When the data are analyzed as in previous studies, not only the Curie temperature T(C) but also the effective critical exponent beta depends upon the terrace width. Taking finite size effects into account gives the terrace width-independent two-dimensional Ising model value beta=1/8.     

Inhomogeneous Ising models with diagonal structure on a square lattice

We study inhomogeneous Ising models on a square lattice. The nearest neighbour couplings are allowed to be of arbitrary strength and sign such that the coupling distribution is translationally invariant in diagonal direction. We calculate the partition function and free energy for a random coupling distribution of finite period. The phase transition is universally of Ising type. The transition temperature is independent of specific details of the coupling distribution. In particular, unexpected results for the absence of a phase transition are derived. Special examples are considered in detail, phase diagrams and critical temperature are determined. We calculate ground state energy and ground state degeneracy or, equivalently, rest entropy for “pure” frustration models, i.e. models with couplings of fixed strength but arbitrary sign, which never show a phase transition at a finite temperature.     

Renormalization group for Markov chains and application to metastability

In this paper we introduce a new renormalization group method for the study of the long-time behavior of Markov chains with finite state space and with transition probabilities exponentially small in an external parameter. A general approach of metastability problems emerges from this analysis and is discussed in detail in the case of a two-dimensional Ising system at low temperature,     

The absence of phase transition for the classical XY-model on Sierpiński pyramid with fractal dimension D=2

For the spin models with continuous symmetry on regular lattices and finite range of interactions, the lower critical dimension is d?=?2. In two dimensions the classical XY-model displays Berezinskii–Kosterlitz–Thouless (BKT) transition associated with unbinding of topological defects (vortices and antivortices). We perform a Monte Carlo study of the classical XY-model on Sierpiński pyramids (SPs) whose fractal dimension is D = log?4/log?2?=?2 and the average coordination number per site is ≈ 7. The specific heat does not depend on the system size which indicates the absence of a long-range order. From the dependence of the helicity modulus on the cluster size and on boundary conditions, we draw a conclusion that in the thermodynamic limit there is no BKT transition at any finite temperature. This conclusion is also supported by our results for linear magnetic susceptibility. The lack of finite temperature phase transition is presumably caused by the finite order of ramification of SP.     

Corner wetting in a far-from-equilibrium magnetic growth model

The irreversible growth of magnetic films is studied in three-dimensional confined geometries of size L×L×M, where M≫L is the growing direction. Competing surface magnetic fields, applied to opposite corners of the growing system, lead to the observation of a localization-delocalization (weakly rounded) transition of the interface between domains of up and down spins on the planes transverse to the growing direction. This effective transition is the precursor of a true far-from-equilibrium corner wetting transition that takes place in the thermodynamic limit. The phenomenon is characterized quantitatively by drawing a magnetic field-temperature phase diagram, firstly for a confined sample of finite size, and then by extrapolating results, obtained with samples of different size, to the thermodynamic limit. The results of this work are a nonequilibrium realization of analogous phenomena recently investigated in equilibrium systems, such as corner wetting transitions in the Ising model.     

Wavevector-Dependent Susceptibility in Quasiperiodic Ising Models

Using the various functional relations for correlation functions in planar Ising models, new results are obtained for the correlation functions and the q-dependent susceptibility for Ising models on a quadratic lattice with quasiperiodic coupling constants. The effects are clearest if the interactions are both attractive and repulsive according to a quasiperiodic pattern. In particular, an exact scaling limit result for the two-point correlation function of the Z-invariant inhomogeneous Ising model is presented and the q-dependent susceptibility is calculated for some cases where the coupling constants vary according to Fibonacci rules. It is found that the ferromagnetic case differs drastically from the case with both ferro- and antiferromagnetic bonds. In the mixed case, the peaks of the q-dependent susceptibility are everywhere dense for temperature T both above or below the critical temperature T_c, but due to overlap only a finite number of peaks is visible. This number of visible peaks decreases as T moves away from T_c. In the ferromagnetic case, there is typically only one single peak at q=0, in spite of the aperiodicity present in the lattice. These results provide evidence that in real systems, even if the atoms arrange themselves aperiodically, there will be no dramatic difference in the diffraction pattern, unless the pair correlation function has clear aperiodic oscillations. The number of oscillations per correlation length determines the number of visible peaks.     

Finite size effects around pseudo-transition in one-dimensional models with nearest neighbor interaction

Recently gigantic peaks in thermodynamic response functions have been observed at finite temperature for one-dimensional models with short-range coupling, closely resembling a second-order phase transition. Thus, we will analyze the finite temperature pseudo-transition property observed in some one-dimensional models and its relationship with finite size effect. In particular, we consider two chain models to study the finite size effects; these are the Ising-Heisenberg tetrahedral chain and an Ising-Heisenberg-type ladder model. Although the anomalous peaks of these one-dimensional models have already been studied in the thermodynamic limit, here we will discuss the finite size effects of the chain and why the peaks do not diverge in the thermodynamic limit. So, we discuss the dependence of the finite size effects, for moderately and sufficiently large systems, in which the specific heat and magnetic susceptibility exhibit peculiar rounded towering peaks for a given temperature. This behavior is quite similar to a continuous phase transition, but there is no singularity. For moderately large systems, the peaks narrow and increase in height as the number of unit cells is increased, and the location of peak shifts slightly. Hence, one can naively induce that the sharp peak should lead to a divergence in the thermodynamic limit. However, for a rather large system, the height of a peak goes asymptotically to a finite value. Our result rigorously confirms the dependence of the peak height with the number of unit cells at the pseudo-critical temperature. We also provide an alternative empirical function that satisfactorily fits specific heat and magnetic susceptibility at pseudo-critical temperature. Certainly, our result is crucial to understand the finite size correction behavior in quantum spin models, which in general are only numerically tractable within the framework of the finite size analysis.     

Monte Carlo study of the two dimensional quadratic ising ferromagnet with mixed spins ofS=1/2 andS=1

The single-spin-flip Metropolis algorithm is applied to an Ising ferromagnet with mixed spins ofS=1/2 andS=1 on the square lattice. The critical temperature obtained from our Monte Carlo simulation is very close to the high temperature series expansion result. The finite size scaling results for the exponents yield the two dimensional Ising values, which are in good agreement with those suggested by the universality hypothesis.