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.     

Scaling Relations for Two-Dimensional Ising Percolation

We consider the percolation problem in the high-temperature Ising model on the two-dimensional square lattice at or near critical external fields. We show that all scaling relations, except for a single hyperscaling relation, hold under the power law assumptions for the one-arm path and the four-arm paths.     

A Short Note on the Scaling Function Constant Problem in the Two-Dimensional Ising Model

We provide a simple derivation of the constant factor in the short-distance asymptotics of the tau-function associated with the 2-point function of the two-dimensional Ising model. This factor was first computed by Tracy (Commun Math Phys 142:297–311, 1991) via an exponential series expansion of the correlation function. Further simplifications in the analysis are due to Tracy and Widom (Commun Math Phys 190:697–721, 1998) using Fredholm determinant representations of the correlation function and Wiener–Hopf approximation results for the underlying resolvent operator. Our method relies on an action integral representation of the tau-function and asymptotic results for the underlying Painlevé-III transcendent from McCoy et al. (J Math Phys 18:1058–1092, 1977).     

Susceptibility of the rectangular Ising ferromagnet

The critical index values= 7/4 for the susceptibility and=15 for the critical isotherm are derived rigorously for the rectangular Ising ferromagnet with nearest neighbor interactions. The critical indices associated with the Fisher moment definition of the correlation length are obtained asTT _c+. The index of the fluctuation sum definition of critical correlations is obtained.Partially supported by grant PHY 76 17191.     

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.     

The Finite-Size Scaling Study of the Ising Model for the Fractals

The fractals are obtained by using the model of diffusion-limited aggregation (DLA) for 40 ≤ L ≤ 240. The two-dimensional Ising model is simulated on the Creutz cellular automaton for 40 ≤ L ≤ 240. The critical exponents and the fractal dimensions are computed to be β = 0.124(8), γ = 1.747(10), α = 0.081(21), δ = 14.994(11), η = 0.178(10), ν = 0.960(23) and \(d_{f}^{\beta } =1.876(8), \,d_{f}^{\gamma } =3.747(10), \,d_{f}^{\alpha } =2.081(68), \,d_{f}^{\delta } =1.940(22)\), \(d_{f}^{\eta } =2.178(10)\), \(d_{f}^{\nu } =2.960(22)\), which are consistent with the theoretical values of β = 0.125, γ = 1.75, α = 0, δ = 15, η = 0.25, ν = 1 and \(d_{f}^{\beta } =1.875, \,d_{f}^{\gamma } =3.75, \,d_{f}^{\alpha } =2, \,d_{f}^{\delta } =1.933, \,d_{f}^{\eta } =2.25, \,d_{f}^{\nu } =3\).     

Scaling and front dynamics in Ising quantum chains

We study the relaxation behaviour of the quantum Ising chain, focusing our attention onto the non-equilibrium dynamics of the transverse magnetization. The initial states, from which the magnetization relaxes, are product states such as those generated by setting in contact several systems, each initially equilibrated at a given temperature. Due to the free fermionic structure of the chain, the dynamics of the transverse magnetization is easily expressed in a compact form. In the completely factorized initial state, corresponding to a situation where all the spins are thermalized independently, we obtain in the scaling limit the Green function associated to the transverse magnetization. The relaxation behaviour is also considered in the system-bath case, where part of the chain called the system is thermalized at a temperature T_s and the remaining part is at a temperature T_b. The magnetization profiles show a scaling behaviour. Moreover, in the extreme case T_b=∞ and T_s=0, it is shown that the magnetization relaxes in quantized steps in the strong transverse field region.     

The Susceptibility of the Square Lattice Ising Model: New Developments

We have made substantial advances in elucidating the properties of the susceptibility of the square lattice Ising model. We discuss its analyticity properties, certain closed form expressions for subsets of the coefficients, and give an algorithm of complexity O(N⁶) to determine its first N coefficients. As a result, we have generated and analyzed series with more than 300 terms in both the high- and low-temperature regime. We quantify the effect of irrelevant variables to the scaling-amplitude functions. In particular, we find and quantify the breakdown of simple scaling, in the absence of irrelevant scaling fields, arising first at order |T–T_c|^9/4, though high-low temperature symmetry is still preserved. At terms of order |T–T_c|^17/4 and beyond, this symmetry is no longer present. The short-distance terms are shown to have the form (T–T_c)^p (log |T–T_c|)^q with pq². Conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments.     

Scaling and self-averaging in the three-dimensional random-field Ising model

We investigate, by means of extensive Monte Carlo simulations, the magnetic critical behavior of the three-dimensional bimodal random-field Ising model at the strong disorder regime. We present results in favor of the two-exponent scaling scenario, [`(h)]\bar{\eta} = 2η, where η and [`(h)]\bar{\eta} are the critical exponents describing the power-law decay of the connected and disconnected correlation functions and we illustrate, using various finite-size measures and properly defined noise to signal ratios, the strong violation of self-averaging of the model in the ordered phase.     

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.     

Reduced Fidelity Susceptibility in One-Dimensional Transverse Field Ising Model

We study critical behaviors of the reduced fidelity susceptibility for two neighboring sites in the onedimensional transverse field Ising model. It is found that the divergent behaviors of the susceptibility take the form of square of logarithm, in contrast with the global ground-state fidelity susceptibility which is power divergence. In order to perform a scaling analysis, we take the square root of the susceptibility and determine the scaling exponent analytically and the result is further confirmed by numerical calculations.     

The Finite-Size Scaling Relation for the Order-Parameter Probability Distribution of the Six-Dimensional Ising Model

The six dimensional Ising model with nearest-neighbor pair interactions has been simulated and verified numerically on the Creutz Cellular Automaton by using five bit demons near the infinite-lattice critical temperature with the linear dimensions L=4,6,8,10. The order parameter probability distribution for six dimensional Ising model has been calculated at the critical temperature. The constants of the analytical function have been estimated by fitting to probability function obtained numerically at the finite size critical point.     

Scaling limit of the energy variable for the two-dimensional Ising ferromagnet

The critical point limit law (scaling limit) of the suitably renormalized energy variable is explicitly calculated for the two-dimensional nearest-neighbour Ising cylinder with free edges. It is shown that the renormalization factor has to behave as (2M 2N lnN)^1/2, where 2M denotes the number of rows and 2N the number of columns. By first taking the limitM and thenN, the limit law is proven to be Gaussian.     

Susceptibility Critical Exponent of a 1D Ising Ring-Type Ferromagnetic

Using the Monte Carlo method, critical behavior of the one-dimensional ferromagnetic Ising model has been investigated with allowance for the interaction of the second and third neighbors and four-particle interaction. The obtained results on the critical temperature were compared with the critical temperature of the quasi-one-dimensional Ising magnetic [(СН₃)₃NH] · FeCl₃ · 2H₂O and with the magnitude of the exchange interaction J/k_B = 17.4 K. Within the scope of the finite-dimensional scaling theory, the critical susceptibility exponent has been calculated. It has been shown that values of the susceptibility exponent for the one-dimensional Ising model with periodic boundary conditions are considerably less than the known values of the exponents for three-dimensional systems. The critical susceptibility exponent strongly depends on energy parameters; namely, it decreases with an increase in them.     

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.     

Scaling properties of hadron production in the Ising model for quark-hadron phase transition

Earlier study of quark-hadron phase transition in the Ginzberg-Landau theory is reexamined in the Ising model, so that spatial fluctuations during the transition can be taken into account. Although the dimension of the physical system is 2, as will be argued, bothd=2 andd=4 Ising systems are studied, the latter being theoretically closer to the Ginzberg-Landau theory. The normalized factorial momentsF _q are used to quantify multiplicity fluctuations, and the scaling exponentν is used to characterize the scaling properties. It is found by simulation on the Ising lattice thatν becomes a function of the temperatureT nearT _c . The average value ofν over a range ofT<T _c agrees with the value of 1.3 derived analytically from the Ginzberg-Landau theory. Thus the implications of the mean-field theory are not invalidated by either the introduction of spatial fluctuations or the restriction to a 2D system.