Lattice Ising Model Research Articles

Singularities of n-fold integrals of the Ising class and the theory of elliptic curves

We introduce some multiple integrals that are expected to have the same singularities as the singularities of the n-particle contributions χ(n) to the susceptibility of the square lattice Ising model. We find the Fuchsian linear differential equation satisfied by these multiple integrals for n = 1, 2, 3, 4 and only modulo some primes for n = 5 and 6, thus providing a large set of (possible) new singularities of χ(n). We discuss the singularity structure for these multiple integrals by solving the Landau conditions. We find that the singularities of the associated ODEs identify (up to n = 6) with the leading pinch Landau singularities. The second remarkable obtained feature is that the singularities of the ODEs associated with the multiple integrals reduce to the singularities of the ODEs associated with a finite number of one-dimensional integrals. Among the singularities found, we underline the fact that the quadratic polynomial condition 1 + 3w + 4w2 = 0, that occurs in the linear differential equation of χ(3), actually corresponds to a remarkable property of selected elliptic curves, namely the occurrence of complex multiplication. The interpretation of complex multiplication for elliptic curves as complex fixed points of the selected generators of the renormalization group, namely isogenies of elliptic curves, is sketched. Most of the other singularities occurring in our multiple integrals are not related to complex multiplication situations, suggesting an interpretation in terms of (motivic) mathematical structures beyond the theory of elliptic curves.

Possibility of Fisher renormalization of the critical exponents in an Ising fluid

Using Monte Carlo simulation techniques, we study the ferromagnetic order-disorder phase transition in Ising spin fluids with hard-core Yukawa interaction truncated at various cutoff radii r{c}. We focus our interest on the dependence of critical quantities such as the Binder cumulant and various exponent ratios on the value of r{c}, and on the question whether the Fisher-renormalized exponents expected for such systems can be observed in the simulations. It turns out that the corrections to scaling decaying with a rather small exponent prevent reaching the asymptotic region with the computational power available. Thus, we observe only effective exponents, with different (nonuniversal) values depending on the cutoff radius. The same behavior is also found for the critical Binder cumulant. Nevertheless, an exact investigation of the effective susceptibility exponent gamma{eff} as a function of temperature seems to point towards a Fisher-renormalized value. For two selected cutoff radii, the critical temperature is determined more accurately using, in addition to the cumulant crossing technique, the scanning technique and the shifting technique, taking into account corrections to scaling. Simulations of Ising fluids with constant cutoff radius and varying Yukawa-tail screening lengths lambda also show a nonuniversal dependence of U{c} on lambda. Finally, we have performed simulations of the Ising lattice model with increasing number of couplings which show the expected asymptotic behavior, independent of the range of interactions.

Monte Carlo studies of the cubic lattice mixed-bond Ising model

In this work, we used the Cluster Wolff algorithms to simulate the dynamics of the system. We obtained the thermodynamic quantities such as magnetization and susceptibility. Critical temperatures were estimated at the maximum of the susceptibility and with these values we made the phase diagram, T C versus p (the concentration of magnetic bonds) for different α values, being α the competition parameter. Our results were compared to those obtained by using a new technique in the effective field theory which employs similar probability distribution within the framework of two-site clusters.

Two critical temperatures of the spin-lattice model on donut-shaped surfaces

We have investigated critical behavior of the Ising lattice model assigned on a donut-shaped surface. By means of Monte Carlo simulations, we have found that the model exhibits two distinct critical temperatures at which zero-field magnetic susceptibility show sharp peaks. This peculiarity is attributed to a spatially non-uniform distribution of the Ising spins on the donut surface.

New universality class of the Ising model on curved geometry

We have considered critical properties of the heptagonal and triangular Ising lattice models defined on negatively curved surfaces. Finite-size scaling analyze reveal that several critical exponents take values distinct from those for the two-dimensional Ising model on a flat plane. This deviation originates from non-Euclidean property of the curved surface that leads to an alteration in global symmetries of the mounted system. Besides, it indicates the occurrence of a novel universality class of the Ising model induced by negative surface curvature.

Beyond series expansions: mathematical structures for the susceptibility of the square lattice Ising model

We first study the properties of the Fuchsian ordinary differential equations for the three and four-particle contributions χ(3) and χ(4) of the square lattice Ising model susceptibility. An analysis of some mathematical properties of these Fuchsian differential equations is sketched. For instance, we study the factorization properties of the corresponding linear differential operators, and consider the singularities of the three and four-particle contributions χ(3) and χ(4), versus the singularities of the associated Fuchsian ordinary differential equations, which actually exhibit new ‘‘Landau-like’’ singularities. We sketch the analysis of the corresponding differential Galois groups. In particular we provide a simple, but efficient, method to calculate the so-called ‘‘connection matrices’’ (between two neighboring singularities) and deduce the singular behaviors of χ(3) and χ(4). We provide a set of comments and speculations on the Fuchsian ordinary differential equations associated with the n-particle contributions χ(n) and address the problem of the apparent discrepancy between such a holonomic approach and some scaling results deduced from a Painlevé oriented approach.

Universal features and tail analysis of the order-parameter distribution of the two-dimensional Ising model: An entropic sampling Monte Carlo study

We present a numerical study of the order-parameter probability density function (PDF) of the square Ising model for lattices with linear sizes L = 80 - 140. A recent efficient entropic sampling scheme, combining the Wang-Landau and broad histogram methods and based on the high levels of the Wang-Landau process in dominant energy subspaces is employed. We find that for large lattices there exists a stable window of the scaled order-parameter in which the full ansatz including the pre-exponential factor for the tail regime of the universal PDF is well obeyed. This window is used to estimate the equation of state exponent and to observe the behavior of the universal constants implicit in the functional form of the universal PDF. The probability densities are used to estimate the universal Privman-Fisher coefficient and to investigate whether one could obtain reliable estimates of the universal constants controlling the asymptotic behavior of the tail regime.

Nonequilibrium Critical Relaxation in the 2D Random Field Ising Model

A particular type of nonequilibrium critical dynamics is discussed in this paper, in which the initial state is generated in the square lattice Ising model with random fields of zero mean and variance h. Thus we interpolate between the ordered (h = 0) and disordered (h → ∞) initial states. In particular we consider the square lattice Ising model in which for weak enough random fields the clusters of parallel spins are percolating in the initial state. We aim to study the effect of correlations in the initial state on the nonequilibrium dynamics, so we measure the nonequilibrium relaxation of the magnetization and the autocorrelation function. The relaxation of the magnetization, characterized by the initial slip exponent, θ, is not altered by the percolating nature of the initial state. However, the starting part of the magnetization curve is affected by a cluster dissolution effect leading to a reentrance in time.

Geometric effects on critical behaviours of the Ising model

We investigate the critical behaviour of the two-dimensional Ising model defined on a curved surface with a constant negative curvature. Finite-size scaling analysis reveals that the critical exponents for the zero-field magnetic susceptibility and the correlation length deviate from those for the Ising lattice model on a flat plane. Furthermore, when reducing the effects of boundary spins, the values of the critical exponents tend to those derived from the mean field theory. These findings evidence that the underlying geometric character is responsible for the critical properties the Ising model when the lattice is embedded on negatively curved surfaces.

A Solvable Decorated Ising Lattice Model

A decorated lattice is suggested and the Ising model on it with three kindsof interactions K1, K2, and K3 is studied. Using an equivalenttransformation, the square decorated Ising lattice is transformed into aregular square Ising lattice with nearest-neighbor, next-nearest-neighbor,and four-spin interactions, and the critical fixed point is found atK1 = 0.5769, K2 = −0.0671, and K3 = 0.3428, which determines thecritical temperature of the system. It is also found that this system andthe regular square Ising lattice, and the eight-vertex model belong to thesame universality class.

Square lattice Ising model susceptibility: connection matrices and singular behaviour of χ(3)and χ(4)

We present a simple, but efficient, way to calculate connection matrices between sets of independent local solutions, defined at two neighboring singular points, of Fuchsian differential equations of quite large orders, such as those found for the third and fourth contribution ($\chi^{(3)}$ and $\chi^{(4)}$) to the magnetic susceptibility of square lattice Ising model. We deduce all the critical behaviors of the solutions $\chi^{(3)}$ and $\chi^{(4)}$, as well as the asymptotic behavior of the coefficients in the corresponding series expansions. We confirm that the newly found quadratic number singularities of the Fuchsian ODE associated to $\chi^{(3)}$ are not singularities of the particular solution $\chi^{(3)}$ itself. We use the previous connection matrices to get the exact expressions of all the monodromy matrices of the Fuchsian differential equation for $\chi^{(3)}$ (and $\chi^{(4)}$) expressed in the same basis of solutions. These monodromy matrices are the generators of the differential Galois group of the Fuchsian differential equations for $\chi^{(3)}$ (and $\chi^{(4)}$), whose analysis is just sketched here. As far as the physics implications of the solutions are concerned, we find challenging qualitative differences when comparing the corrections to scaling for the full susceptibillity $\chi$ at high temperature (resp. low temperature) and the first two terms $\chi^{(1)}$ and $\chi^{(3)}$ (resp. $\chi^{(2)}$ and $\chi^{(4)}$) .

Discrete Painlevé equations for a class of PVI τ-functions given as U(N) averages

In a recent work, difference equations (Laguerre–Freud equations) for the bi-orthogonal polynomials and related quantities corresponding to the weight on the unit circle were derived. It is shown here that in the case m = 3, these difference equations, when applied to the calculation of the underlying U(N) average, reduce to a coupled system identifiable with that obtained by Adler and van Moerbeke, using the methods of the Toeplitz lattice and Virasoro constraints. Moreover, it is shown that this coupled system can be reduced to yield the discrete fifth Painlevé equation dPV as it occurs in the theory of the sixth Painlevé system. Methods based on affine Weyl group symmetries of Bäcklund transformations have previously yielded the dPV equation, but with different parameters for the same problem. We find an explicit mapping between the two forms. Applications of our results are made to give recurrences for the gap probabilities and moments in the circular unitary ensemble of random matrices, and to the diagonal spin–spin correlation function of the square lattice Ising model.

Comment on ‘The Fuchsian differential equation of the square lattice Ising model χ(3)susceptibility’

The new singularities in χ(3) found by Zenine et al (2004 J. Phys. A: Math. Gen. 37 9651) are pinch-type singularities analogous to those found in Feynman graphs in field theory. A general formula for the possible location of all such singularities for arbitrary χ(N) is given.

Ising model susceptibility: the Fuchsian differential equation for χ(4)and its factorization properties

We give the Fuchsian linear differential equation satisfied by $\chi^{(4)}$, the ``four-particle'' contribution to the susceptibility of the isotropic square lattice Ising model. This Fuchsian differential equation is deduced from a series expansion method introduced in two previous papers and is applied with some symmetries and tricks specific to $\chi^{(4)}$. The corresponding order ten linear differential operator exhibits a large set of factorization properties. Among these factorizations one is highly remarkable: it corresponds to the fact that the two-particle contribution $\chi^{(2)}$ is actually a solution of this order ten linear differential operator. This result, together with a similar one for the order seven differential operator corresponding to the three-particle contribution, $\chi^{(3)}$, leads us to a conjecture on the structure of all the $ n$-particle contributions $ \chi^{(n)}$.

Square lattice Ising model susceptibility: series expansion method and differential equation for χ(3)

In a previous paper (J. Phys. A {\bf 37} (2004) 9651-9668) we have given the Fuchsian linear differential equation satisfied by $\chi^{(3)}$, the ``three-particle'' contribution to the susceptibility of the isotropic square lattice Ising model. This paper gives the details of the calculations (with some useful tricks and tools) allowing one to obtain long series in polynomial time. The method is based on series expansion in the variables that appear in the $(n-1)$-dimensional integrals representing the $n$-particle contribution to the isotropic square lattice Ising model susceptibility $\chi $. The integration rules are straightforward due to remarkable formulas we derived for these variables. We obtain without any numerical approximation $\chi^{(3)}$ as a fully integrated series in the variable $w=s/2/(1+s^{2})$, where $ s =sh (2K)$, with $K=J/kT$ the conventional Ising model coupling constant. We also give some perspectives and comments on these results.

Spin frustration and antiferromagnetic long range order in [formula omitted] ( [formula omitted] earth)

Antiferromagnetic ordering and spin frustration behaviors in the series of isostructural compounds, R 2 BaPdO 5 ( R = rare earth), are reported. We have investigated the crystallographic, magnetic and thermal properties of the compounds, R 2 BaPdO 5 ( R = Nd , Eu, Gd and Ho). R 2 BaPdO 5 ( R = Nd , Gd and Ho) show paramagnetic properties at high temperatures originating only from the rare-earth ions and magnetic order at 2.1, 2.9 and 4.1 K, respectively. Ho 2BaPdO 5 shows a symmetrical peak of the heat capacity at T N and negligible residual entropy below the T N , suggesting a simple square lattice Ising model. On the contrary, Nd 2BaPdO 5 and Gd 2BaPdO 5 show Schottky type heat capacity at T N and large residual entropy far below the T N , suggesting spin frustration in them. The origin of the possible spin frustration in R 2 BaPdO 5 ( R = Nd and Gd) is discussed in terms of relative strength of J and J ′ in the Shastry–Sutherland spin lattice.

The Fuchsian differential equation of the square lattice Ising model (3)susceptibility

Using an expansion method in the variables $ x_i$ that appear in the $(n-1)$-dimensional integrals representing the $n$-particle contribution to the Ising square lattice model susceptibility $\chi$, we generate a long series of coefficients for the 3-particle contribution $\chi^{(3)}$, using a $ N^4$ polynomial time algorithm. We give the Fuchsian differential equation of order seven for $\chi^{(3)}$ that reproduces all the terms of our long series. An analysis of the properties of this Fuchsian differential equation is performed.

Finite temperature results on the 2D Ising model with mixed perturbation

A numerical study of finite temperature features of thermodynamical observables is performed for the lattice 2d Ising model. Our results support the conjecture that the Finite Size Scaling analysis employed in the study of integrable perturbation of Conformal Field Theory is still valid in the present case, where a non-integrable perturbation is considered.

Classical and Quantum Nonlinear Integrable Systems: Theory and Application

This is a very interesting collection of introductory and review articles on the theory and applications of classical and quantum integrable systems. The book reviews several integrable systems such as the KdV equation, vertex models, RSOS and IRF models, spin chains, integrable differential equations, discrete systems, Ising, Potts and other lattice models and reaction--diffusion processes, as well as outlining major methods of solving integrable systems. These include Lax pairs, Bäcklund and Miura transformations, the inverse scattering method, various types of the Bethe Ansatz, Painlevé methods, the dbar method and fusion methods to mention just a few.

Description of far-from-equilibrium processes by mean-field lattice gas models

Mean-field kinetic equations are a valuable tool to study the atomic dynamics and spin dynamics of simple lattice gas and Ising models. They can be derived from the microscopic master equation of the system and contain analytical expressions for kinetic coefficients and thermodynamic quantities which are usually introduced phenomenologically. We review several methods to obtain such equations, and discuss applications to the dynamics of order–disorder transitions, spinodal decomposition, and dendritic growth in the isothermal or chemical model. In the case of dendritic growth we show that the mean-field kinetic equations are equivalent to standard continuum equations for this problem and derive expressions for macroscopic quantities, e.g. the surface tension and kinetic coefficients, as functions of the microscopic order parameters. In spinodal decomposition, we focus our attention on the vacancy mechanism, which is a more faithful picture of diffusion in solids than the more widely examined exchange mech...