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.     

Linked-Cluster Expansion of the Ising Model

The linked-cluster expansion technique for the high-temperature expansion of spin modes is reviewed. A new algorithm for the computation of three-point and higher Green's functions is presented. Series are computed for all components of two-point Green's functions for a generalized 3D Ising model, to 25th order on the bcc lattice and to 23rd order on the sc lattice. Series for zero-momentum four-, six-, and eight-point functions are computed to 21st, 19th, and 17th order respectively on the bcc lattice.     

The Finite-Size Scaling Functions of the Four-Dimensional Ising Model

A finite-size scaling function of the Privman–Fisher form is proposed for the singular part of the free-energy density of the four-dimensional Ising model. It leads to the finite-size scaling relations available and to the prediction of new ones.     

Susceptibility of the Kagomé lattice Ising model

We explicitly calculate the zero-field magnetic susceptibility of the anisotropic Kagomé lattice Ising model on two different varieties of the parameter space. One of them is the limitH=0 of the solubility condition, obtained in a previous paper by Giacomini, for the model with magnetic field. The other one is the disorder variety of the model, for which a dimensional reduction occurs. These varieties do not contain any nontrivial critical behavior of the model. A functional relation is also established, which relates the zero-field susceptibility for ferromagnetic and competing interactions.     

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.     

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.     

High—Temperature Cutoff Approximation of the 3D kinetic Ising Model

A single-spin transition critical dynamics is used to investigate the three-dimensional kinetic Ising model on an anisotropic cubic lattice,We first derive the fundamental dynamical equations.and then linearize them by a cutoff approximation.We obtain the approximate solutions of the local magnetization and equal-time pair correlation function approximation.We obtain the approximate solutions of the local magnetization and equal-time pair correlation function in zero field.In which the axial-decoupling terms γ1γ2,γ2γ3and γ1γ3as higher infinitesimal quantity are ignored,where γα=tanh(2k0633)=tanh(2Jα/kβT）（α=1,2,3,)We think that it is reasonable as the temperature of the system is very high.The result of what we obtain in this paper can go back to the one-dimensional Glauber‘s theory as long as k2=k3=0.     

The Percolation Transition for the Zero-Temperature Stochastic Ising Model on the Hexagonal Lattice

On the planar hexagonal lattice , we analyze the Markov process whose state (t), in , updates each site v asynchronously in continuous time t0, so that _v (t) agrees with a majority of its (three) neighbors. The initial _v (0)'s are i.i.d. with P[ _v (0)=+1]=p[0,1]. We study, both rigorously and by Monte Carlo simulation, the existence and nature of the percolation transition as t and p1/2. Denoting by ⁺(t,p) the expected size of the plus cluster containing the origin, we (1) prove that ⁺(,1/2)= and (2) study numerically critical exponents associated with the divergence of ⁺(,p) as p1/2. A detailed finite-size scaling analysis suggests that the exponents and of this t= (dependent) percolation model have the same values, 4/3 and 43/18, as standard two-dimensional independent percolation. We also present numerical evidence that the rate at which (t)() as t is exponential.     

The analytic structure of lattice models — why can’t we solve most models?

We investigate the solvability of a variety of well-known problems in lattice statistical mechanics. We provide a new numerical procedure which enables one to conjecture whether the solution falls into a class of functions calleddifferentiably finite functions. Almost all solved problems fall into this class. The fact that one can conjecture whether a given problem is or is not D-finite then informs one as to whether the solution is likely to be tractable or not. We also show how, for certain problems, it is possible to prove that the solutions are notD-finite, based on the work of Rechnitzer [1–3].     

Analyticity properties and a convergent expansion for the inverse correlation length of the low-temperatured-dimensional Ising model

We show that the inverse correlation lengthm(z) of the truncated spin-spin correlation function of theZ ^d Ising model with + or — boundary conditions admits the representationm(z) = –(4d–4)ln z(1–_d1) + r(z) for smallz=e ^–, i.e., large inverse temperatures is ad-dependent analytic function atz = 0, already known in closed form ford = 1 and 2; ford = 3 b_n can be computed explicitly from a finite number of the Z^d limits of z = 0 Taylor series coefficients of the finite lattice correlation function at a finite number of points ofZ ^d.     

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.     

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.     

The finite-size scaling study of the specific heat and the Binder parameter for the six-dimensional Ising model

The six-dimensional Ising model with nearest-neighbor pair interactions is simulated on the Creutz cellular automaton by using the finite-size lattices with the linear dimensions L=4,6,8,10. The temperature variations and the finite-size scaling plots of the specific heat and Binder parameter verify the theoretically predicted expression near the infinite lattice critical temperature. The approximate values for the critical temperature of the infinite lattice T_C=10.838(1), T_C=10.836(20) and T_C=10.835(1) are obtained from the intersection points of specific heat curves, Binder parameter curves and the straight line fit of specific heat maxima, respectively. These results are in agreement with the more precise value of T_C=10.835(5). The value obtained for the critical exponent of the specific heat, i.e., =0.012(2) is also in agreement with =0 predicted by the theory.     

Thermodynamic Transitions of Antiferromagnetic Ising Model on the Fractional Multi-branched Husimi Recursive Lattice

The multi-branched Husimi recursive lattice is extended to a virtual structure with fractional numbers of branches joined on one site. Although the lattice is undrawable in real space, the concept is consistent with regular Husimi lattice. The Ising spins of antiferromagnetic interaction on such a set of lattices are calculated to check the critical temperatures (T_c) and ideal glass transition temperatures (T_k) variation with fractional branch numbers. Besides the similar results of two solutions representing the stable state (crystal) and metastable state (supercooled liquid) and indicating the phase transition temperatures, the phase transitions show a well-defined shift with branch number variation. Therefore the fractional branch number as a parameter can be used as an adjusting tool in constructing a recursive lattice model to describe real systems.     

Two-Point Correlations and Critical Line of the Driven Ising Lattice Gas in a High-Temperature Expansion

Based on a high-temperature expansion, we compute the two-point correlation function and the critical line of an Ising lattice gas driven into a nonequilibrium steady state by a uniform bias E. The lowest nontrivial order already reproduces the key features, i.e., the discontinuity singularity of the structure factor and the (qualitative) E dependence of the critical line. Our approach is easily generalized to other nonequilibrium lattice models and provides a simple analytic tool for the study of the high-temperature phase and its boundaries.     

The roughening transition of the three-dimensional Ising interface: A Monte Carlo study

We study the roughening transition of an interface in an Ising system on a 3D simple cubic lattice using a finite-size scaling method. The particular method has recently been proposed and successfully tested for various solid-on-solid models. The basic idea is the matching of the renormalization-groupflow of the interface with that of the exactly solvable body-centered cubic solid-on-solid model. We unambiguously confirm the Kosterlitz-Thouless nature of the roughening transition of the Ising interface. Our result for the inverse transition temperatureK _r=0.40754(5) is almost two orders of magnitude more accurate than the estimate of Mon, Landau, and Stauffer.     

Weak Singularities of the Isothermal Entropy Change as the Smoking Gun Evidence of Phase Transitions of Mixed-Spin Ising Model on a Decorated Square Lattice in Transverse Field

The magnetocaloric response of the mixed spin-1/2 and spin-S (S>1/2) Ising model on a decorated square lattice is thoroughly examined in presence of the transverse magnetic field within the generalized decoration-iteration transformation, which provides an exact mapping relation with an effective spin-1/2 Ising model on a square lattice in a zero magnetic field. Temperature dependencies of the entropy and isothermal entropy change exhibit an outstanding singular behavior in a close neighborhood of temperature-driven continuous phase transitions, which can be additionally tuned by the applied transverse magnetic field. While temperature variations of the entropy display in proximity of the critical temperature Tc a striking energy-type singularity (T−Tc)log|T−Tc|, two analogous weak singularities can be encountered in the temperature dependence of the isothermal entropy change. The basic magnetocaloric measurement of the isothermal entropy change may accordingly afford the smoking gun evidence of continuous phase transitions. It is shown that the investigated model predominantly displays the conventional magnetocaloric effect with exception of a small range of moderate temperatures, which contrarily promotes the inverse magnetocaloric effect. It turns out that the temperature range inherent to the inverse magnetocaloric effect is gradually suppressed upon increasing of the spin magnitude S.     

Distribution of Avalanche Sizes in the Hysteretic Response of the Random-Field Ising Model on a Bethe Lattice at Zero Temperature

We consider the zero-temperature single-spin-flip dynamics of the random-field Ising model on a Bethe lattice in the presence of an external field h. We derive the exact self-consistent equations to determine the distribution Prob(s) of avalanche sizes s as the external field increases from – to . We solve these equations explicitly for a rectangular distribution of the random fields for a linear chain and the Bethe lattice of coordination number z=3, and show that in these cases, Prob(s) decreases exponentially with s for large s for all h on the hysteresis loop. We find that for z4 and for small disorder, the magnetization shows a first-order discontinuity for several continuous and unimodal distributions of the random fields. The avalanche distribution Prob(s) varies as s ^–3/2 for large s near the discontinuity.     

Wavevector-Dependent Susceptibility in <Emphasis Type="Italic">Z</Emphasis>-Invariant Pentagrid Ising Model

We study the q-dependent susceptibility χ(q) of a Z-invariant ferromagnetic Ising model on a Penrose tiling, as first introduced by Korepin using de Bruijn's pentagrid for the rapidity lines. The pair-correlation function for this model can be calculated exactly using the quadratic difference equations from our previous papers. Its Fourier transform χ(q) is studied using a novel way to calculate the joint probability for the pentagrid neighborhoods of the two spins, reducing this calculation to linear programming. Since the lattice is quasiperiodic, we find that χ(q) is aperiodic and has everywhere dense peaks, which are not all visible at very low or high temperatures. More and more peaks become visible as the correlation length increases—that is, as the temperature approaches the critical temperature. Supported in part by NSF Grant No. PHY 01-00041.     

The role of dimensionality in the kinetic Ising model of spinodal decomposition: Evidence from zero-temperature quenches

We study a three-dimensional Ising lattice gas model with spin-exchange dynamics quenched from infinite to zero temperature. We consider a wide range of values of the binary composition (i.e., magnetization) and annealed vacancy concentration. We find that, as in two dimensions, the system freezes in a configuration very far from equilibrium, and that the interface energy per bond in the frozen state, which is very large, in all cases takes very nearly the same values as in two dimensions. We discuss the implications of these results regarding the irrelevance of dimensionality in this problem.