A Renormalization Group Study of the Ising Model on a Triangular Lattice with Long-Range Interactions

The critical exponents of the triangular lattice Ising model with long-range interactions γ^-s are calculated by the real space renormalization group. Using the simplest Kadanoff blocks and the lowest approximation of cumulant expansion, it is shown that there exists a finite critical temperature when 4(1 - ㏑2/㏑3) < s < 4.     

Analytical Expressions for Ising Models on High Dimensional Lattices

We use an m-vicinity method to examine Ising models on hypercube lattices of high dimensions d≥3. This method is applicable for both short-range and long-range interactions. We introduce a small parameter, which determines whether the method can be used when calculating the free energy. When we account for interaction with the nearest neighbors only, the value of this parameter depends on the dimension of the lattice d. We obtain an expression for the critical temperature in terms of the interaction constants that is in a good agreement with the results of computer simulations. For d=5,6,7, our theoretical estimates match the numerical results both qualitatively and quantitatively. For d=3,4, our method is sufficiently accurate for the calculation of the critical temperatures; however, it predicts a finite jump of the heat capacity at the critical point. In the case of the three-dimensional lattice (d=3), this contradicts the commonly accepted ideas of the type of the singularity at the critical point. For the four-dimensional lattice (d=4), the character of the singularity is under current discussion. For the dimensions d=1, 2 the m-vicinity method is not applicable.     

Finite-Size Scaling for the 2D Ising Model with Minus Boundary Conditions

We study the magnetization m _L (h, ) for the Ising model on a large but finite lattice square under the minus boundary conditions. Using known large-deviation results evaluating the balance between the competing effects of the minus boundary conditions and the external magnetic field h, we describe the details of its dependence on h as exemplified by the finite-size rounding of the infinite-volume magnetization discontinuity and its shift with respect to the infinite-volume transition point.     

Tests for Monte Carlo renormalization studies on Ising models

In this expanded version of an earlier letter, we consider many computational details that were omitted for want of space. Ford = 2 Ising spins with up to 13 different short-range interactions, we construct the critical surface in the vicinity of (Onsager's) nearest-neighbor (nn) critical point by using the body of the available information on the solvable nn case. We then see if the Monte Carlo renormalization group flows generated from this point do indeed lie on this surface and quantify the errors if they do not.     

New Results for the Correlation Functions of the Ising Model and the Transverse Ising Chain

In this paper we show how an infinite system of coupled Toda-type nonlinear differential equations derived by one of us can be used efficiently to calculate the time-dependent pair-correlations in the Ising chain in a transverse field. The results are seen to match extremely well long large-time asymptotic expansions newly derived here. For our initial conditions we use new long asymptotic expansions for the equal-time pair correlation functions of the transverse Ising chain, extending an old result of T.T. Wu for the 2d Ising model. Using this one can also study the equal-time wavevector-dependent correlation function of the quantum chain, a.k.a. the q-dependent diagonal susceptibility in the 2d Ising model, in great detail with very little computational effort. Supported in part by the National Science Foundation under grant PHY 07-58139 and by the Australian Research Council under Project ID: LX0989627.     

Ising Model on an Infinite Ladder Lattice

In this paper we propose an Ising model on an infinite ladder lattice, which is made of two infinite Ising spin chains with interactions. It is essentially a quasi-one-dimessional Ising model because the length of the ladder lattice is infinite, while its width is finite. We investigate the phase transition and dynamic behavior of Ising model on this quasi-one-dimessional system. We use the generalized transfer matrix method to investigate the phase transition of the system. It is found that there is no nonzero temperature phase transition in this system. At the same time, we are interested in Glauber dynamics. Based on that, we obtain the time evolution of the local spin magnetization by exactly solving a set of master equations.     

Distribution of Energies for the Two-Dimensional Ising Model

We calculate the number of polygons with fixed total length drawn on a square lattice with periodic boundary conditions. In addition, we study the statistics of polygons with the number of horizontal and vertical links fixed separately. The analysis is performed via a mapping to the Ising model with isotropic and anisotropic interactions. We deal with the case of finite lattice sizes as well as the thermodynamic limit.     

Poisson Approximations for the Ising Model

A d-dimensional Ising model on a lattice torus is considered. As the size n of the lattice tends to infinity, a Poisson approximation is given for the distribution of the number of copies in the lattice of any given local configuration, provided the magnetic field a = a(n) tends to −∞ and the pair potential b remains fixed. Using the Stein-Chen method, a bound is given for the total variation error in the ferromagnetic case. AMS SUBJECT CLASSIFICATION: 60F05, 82B20.     

Ising Model on an Infinite Ladder Lattice

In this paper we propose an Ising model on an infinite ladder lattice, which is made of two infinite Ising spin chains with interactions. It is essentially a quasi-one-dimessional Ising model because the length of the ladder lattice is infinite, while its width is finite. We investigate the phase transition and dynamic behavior of Ising model on this quasi-one-dimessional system. We use the generalized transfer matrix method to investigate the phase transition of the system. It is found that there is no nonzero temperature phase transition in this system. At the same time, we are interested in Glauber dynamics. Based on that, we obtain the time evolution of the local spin magnetization by exactly solving a set of master equations.     

Critical Value of the Quantum Ising Model on Star-Like Graphs

We present a rigorous determination of the critical value of the ground-state quantum Ising model in a transverse field, on a class of planar graphs which we call star-like. These include the junction of several copies of ℤ at a single point. Our approach is to use the graphical, or fk-, representation of the model, and the probabilistic and geometric tools associated with it. This research was carried out during the author’s Ph.D. studentship at the University of Cambridge, UK, and the Royal Institute of Technology (KTH), Sweden. The author gratefully acknowledges funding from KTH during this period. The author would also like to thank Riddarhuset, Stockholm, for generous support during his studies.     

Effective-Field Theory for Kinetic Ising Model on Honeycomb Lattice

As an analytical method, the effective-field theory （EFT） is used to study the dynamical response of the kinetic Ising model in the presence of a sinusoidal oscillating field. The effective-field equations of motion of the average magnetization are given for the honeycomb lattice （Z = 3）. The Liapunov exponent A is calculated for discussing the stability of the magnetization and it is used to determine the phase boundary. In the field amplitude ho / ZJ-temperature T/ZJ plane, the phase boundary separating the dynamic ordered and the disordered phase has been drawn. In contrast to previous analytical results that predicted a tricritical point separating a dynamic phase boundary line of continuous and discontinuous transitions, we find that the transition is always continuous. There is inconsistency between our results and previous analytical results, because they do not introduce sufficiently strong fluctuations.     

The distribution of clusters for the Ising model

We rigorously prove that the probabilityP _n for the origin to belong to a cluster of exactlyn positive spins in thev-dimensional Ising model behaves as exp(–n^{(v – 1)/v}) in various regions, including in particular the low-temperature positive and negative phases in zero magnetic field.     

Density of the Fisher Zeroes for the Ising Model

The density of the Fisher zeroes, or zeroes of the partition function in the complex temperature plane, is determined for the Ising model in zero field as well as in a pure imaginary field i/2. Results are given for the simple-quartic, triangular, honeycomb, and the kagomé lattices. It is found that the density diverges logarithmically at points along its loci in appropriate variables.     

Exact solution on the magnetism and ferroelectricity in the Ising spin chain

In this paper, we explore the magnetoelectric coupling in Ca₃CoMnO₆-type compound with the consideration of interaction between spins. Under the linear approximation of nearest-neighbor spin interaction with respect to the ion displacement, both the Hamiltonian and the partition function of the system can be simplified as the summation of two independent items, one is linear harmonic oscillator relevant to the lattice vibration, and the other is only relevant to the spin. We obtain the magnetic and ferroelectric quantities of Ca₃CoMnO₆-type rigorously on the basis of the transfer-matrix method, qualitatively exhibiting the corresponding curves taken in the presence of temperature for zero magnetic field and various magnetic fields, respectively, and our calculation results are basically consistent with the behaviors in the experiment. We find that the magnetic susceptibility in the absence of magnetic field takes on the features of ferromagnetic Ising-like behavior. Moreover, the influence of different next-nearest-neighbor exchange interaction on the magnetic susceptibility and relative dielectric constant is given as well, exhibiting the corresponding complex response of the magnetic susceptibility based on the up-up-down-down spin structure.     

Mean-Field Behavior for Long- and Finite Range Ising Model,Percolation and Self-Avoiding Walk

We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if d>2(α ∧2) for self-avoiding walk and the Ising model, and d>3(α ∧2) for percolation, where d denotes the dimension and α the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (Ann. Probab. 33(5):1886–1944, 2005).      

Surface tension from finite-volume vacuum tunneling in the 3D Ising model

We measure the surface tension in the broken phase of the 3D Ising model at a temperatureT=0.955T _c with two different methods which are taken from quantum field theory in finite volumes. Both methods rely on finite-size effects close to the phase transition. The first one measures from the size dependence of the vacuum tunneling energy, which is determined by the decay of a correlation, giving=0.030. The second one extracts from the size dependence of the rate of flip events and its corresponding correlation time. It leads to=0.027. Both values agree reasonably with other calculations.     

An Improved Heterogeneous Mean-Field Theory for the Ising Model on Complex Networks

Heterogeneous mean-field theory is commonly used methodology to study dynamical processes on complex networks,such as epidemic spreading and phase transitions in spin models.In this paper,we propose an improved heterogeneous mean-field theory for studying the Ising model on complex networks.Our method shows a more accurate prediction in the critical temperature of the Ising model than the previous heterogeneous mean-field theory.The theoretical results are validated by extensive Monte Carlo simulations in various types of networks.     

An On-Line Library of Extended High-Temperature Expansions of Basic Observables for the Spin-S Ising Models on Two- and Three-Dimensional Lattices

We present an on-line library of unprecedented extension for high-temperature expansions of basic observables in the Ising models of general spin S, with nearest-neighbor interactions. We have tabulated through order ²⁵ the series for the nearest-neighbor correlation function, the susceptibility and the second correlation moment in two dimensions on the square lattice, and, in three dimensions, on the simple-cubic and the body-centered cubic lattices. The expansion of the second field derivative of the susceptibility is also tabulated through ²³ for the same lattices. We have thus added several terms (from four up to thirteen) to the series already published for spin S = 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 5, .     

On Criticality for Competing Influences of Boundary and External Field in the Ising Model

We continue a study of Schonmann (1994), Schonmann and Shlosman (1996), and Greenwood and Sun (1997) regarding the competing influences of boundary conditions and external field for the Ising model. We find a critical point B ₀ in the competing influences for low temperature in dimension d 2A7E; 2.