Ising Cluster Kinetics at the Critical Point

The scaling laws of Edwards et al. for cluster fragmentation at the two-dimensional percolation threshold, recently confirmed also in high dimensions, explain the Alexandrowicz observation that the Becker–Döring equation of classical nucleation theory and its generalization by Katz, Saltsburg, and Reiss fail right at the critical point.     

The Critical Ising Model via Kac-Ward Matrices

The Kac-Ward formula allows to compute the Ising partition function on any finite graph G from the determinant of 2^2g matrices, where g is the genus of a surface in which G embeds. We show that in the case of isoradially embedded graphs with critical weights, these determinants have quite remarkable properties. First of all, they satisfy some generalized Kramers-Wannier duality: there is an explicit equality relating the determinants associated to a graph and to its dual graph. Also, they are proportional to the determinants of the discrete critical Laplacians on the graph G, exactly when the genus g is zero or one. Finally, they share several formal properties with the Ray-Singer ${\overline{\partial}}$ -torsions of the Riemann surface in which G embeds.     

Critical Point of the Honeycomb Antiferromagnetic Ising Model in a Nonzero Magnetic Field: An Analytical Analysis

In this paper we determined the critical point of the antiferromagnetic Ising model in a nonzero magnetic field on the honeycomb lattice by an analytical method-the generalized cumulant expansion with the mean field hypothesis. By calculating the magnetization to the fourth order correction, we get encouraging results when compared with the known numerical results by finite size analysis.     

Non-self-averaging in the Critical Point of a Random Ising Ferromagnet

Journal of Experimental and Theoretical Physics - In this paper, we review recent results on sample-to-sample fluctuations in a critical Ising model with quenched random ferromagnetic couplings. In...     

Holomorphic Spinor Observables in the Critical Ising Model

We introduce a new version of discrete holomorphic observables for the critical planar Ising model. These observables are holomorphic spinors defined on double covers of the original multiply connected domain. We compute their scaling limits, and show their relation to the ratios of spin correlations, thus providing a rigorous proof to a number of formulae for those ratios predicted by CFT arguments.     

Crossover Critical Behavior in the Three-Dimensional Ising Model

The character of critical behavior in physical systems depends on the range of interactions. In the limit of infinite range of the interactions, systems will exhibit mean-field critical behavior, i.e., critical behavior not affected by fluctuations of the order parameter. If the interaction range is finite, the critical behavior asymptotically close to the critical point is determined by fluctuations and the actual critical behavior depends on the particular universality class. A variety of systems, including fluids and anisotropic ferromagnets, belongs to the three-dimensional Ising universality class. Recent numerical studies of Ising models with different interaction ranges have revealed a spectacular crossover between the asymptotic fluctuation-induced critical behavior and mean-field-type critical behavior. In this work, we compare these numerical results with a crossover Landau model based on renormalization-group matching. For this purpose we consider an application of the crossover Landau model to the three-dimensional Ising model without fitting to any adjustable parameters. The crossover behavior of the critical susceptibility and of the order parameter is analyzed over a broad range (ten orders) of the scaled distance to the critical temperature. The dependence of the coupling constant on the interaction range, governing the crossover critical behavior, is discussed.     

Vanishing Critical Magnetization in the Quantum Ising Model

Adapting the recent argument of Aizenman, Duminil-Copin and Sidoravicius for the classical Ising model, it is shown here that the magnetization in the transverse-field Ising model vanishes at the critical point. The proof applies to the ground state in dimension d ≥ 2 and to positive-temperature states in dimension d ≥ 3, and relies on graphical representations as well as an infrared bound.     

Geometric Upper Critical Dimensions of the Ising Model

The upper critical dimension of the Ising model is known to be dc = 4, above which critical behavior is regarded to be trivial. We hereby argue from extensive simulations that, in the random-cluster representation,the Ising model simultaneously exhibits two upper critical dimensions at(d_c = 4, d_p = 6), and critical clusters for d ≥ d_p, except the largest one, are governed by exponents from percolation universality. We predict a rich variety of geometric propertie...     

Stochastic Quantization Approach for the Ising Model

The Ising model is studied in the fermionicformulation of the stochastic quantization. An exactstochastic equation is given for D = 2 and 3 and in aHartree approximation a method is developed for treating the two-point correlation functions.     

The Critical Behavior of the Ising Model on Union Jack Lattice

In this paper, we have investigated the critical behavior of the ferromagnetic Ising model on union jack lattice. The model is equivalent to the eight-vertex model which can be solved as a free fermion model with the free fermion approximation. The critical exponents have been obtained as aα = α = 0, β = β'= 0.125, γ = γ' = 0.875 and δ = δ' = 8.     

From Cycle Rooted Spanning Forests to the Critical Ising Model: an Explicit Construction

Fisher established an explicit correspondence between the 2-dimensional Ising model defined on a graph G and the dimer model defined on a decorated version ${\mathcal{G}}$ of this graph (Fisher in J Math Phys 7:1776–1781, 1966). In this paper we explicitly relate the dimer model associated to the critical Ising model and critical cycle rooted spanning forests (CRSFs). This relation is established through characteristic polynomials, whose definition only depends on the respective fundamental domains, and which encode the combinatorics of the model. We first show a matrix-tree type theorem establishing that the dimer characteristic polynomial counts CRSFs of the decorated fundamental domain ${\mathcal{G}_1}$ . Our main result consists in explicitly constructing CRSFs of ${\mathcal{G}_1}$ counted by the dimer characteristic polynomial, from CRSFs of G ₁, where edges are assigned Kenyon’s critical weight function (Kenyon in Invent Math 150(2):409–439, 2002); thus proving a relation on the level of configurations between two well known 2-dimensional critical models.     

The Ising Model on Pure Husimi Lattices: A General Formulation and the Critical Temperatures

We consider the Ising spin 1/2 model on arbitrary pure Husimi lattices. An effective representation for the recursion relations is found which allows to write the general solution of the model in an fluent unified way for all pure Husimi lattices. In this respect, explicit expressions for the spontaneous magnetization, for the susceptibility, for the free energy, and for the specific heat are found. Besides, it is shown that this representation allows also to determine exactly the position of the critical temperature on arbitrary pure Husimi lattice. It is found that the critical temperatures for all pure Husimi lattices are driven by a single polynomial equation with coefficients given by parameters that uniquely describe the lattices.     

The Critical Behavior of the Antiferromagnetic Ising Model with Long-Range Interaction Effects

In this paper, the antiferromagnetic Ising model with ferromagnetic long-range interaction is modeled by the Monte Carlo method. The case of ferromagnetic long-range forces decreasing by a power law is considered. The dependence of the phase-transition temperature on long-range interaction parameters is obtained. The phase diagram was constructed at different values of long-range interaction parameters. Conditions for the existence of the frustrated state of the system were revealed.     

The High Temperature Ising Model on the Triangular Lattice is a Critical Bernoulli Percolation Model

We define a new percolation model by generalising the FK representation of the Ising model, and show that on the triangular lattice and at high temperatures, the critical point in the new model corresponds to the Ising model. Since the new model can be viewed as Bernoulli percolation on a random graph, our result makes an explicit connection between Ising percolation and critical Bernoulli percolation, and gives a new justification of the conjecture that the high temperature Ising model on the triangular lattice is in the same universality class as Bernoulli percolation.     

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.     

Critical Region for Droplet Formation in the Two-Dimensional Ising Model

We study the formation/dissolution of equilibrium droplets in finite systems at parameters corresponding to phase coexistence. Specifically, we consider the 2D Ising model in volumes of size L ² , inverse temperature > _c and overall magnetization conditioned to take the value m L ² –2m v _L , where _c ^–1 is the critical temperature, m =m () is the spontaneous magnetization and v _L is a sequence of positive numbers. We find that the critical scaling for droplet formation/dissolution is when v _L ^3/2 L ^–2 tends to a definite limit. Specifically, we identify a dimensionless parameter , proportional to this limit, a non-trivial critical value _c and a function such that the following holds: For < _c , there are no droplets beyond log L scale, while for > _c , there is a single, Wulff-shaped droplet containing a fraction _c =2/3 of the magnetization deficit and there are no other droplets beyond the scale of log L. Moreover, and are related via a universal equation that apparently is independent of the details of the system.     

Phase Transition for the Ising Model on the Critical Lorentzian Triangulation

The ferromagnetic Ising model without external field on an infinite Lorentzian triangulation sampled from the uniform distribution is considered. We prove uniqueness of the Gibbs measure in the high temperature region and coexistence of at least two Gibbs measures at low temperature. The proofs are based on the disagreement percolation method and on a variant of the Peierls contour method. The critical temperature is shown to be constant a.s.     

Dimers and the Critical Ising Model on lattices of genus >1

We study the partition function of both Close-Packed Dimers and the Critical Ising Model on a square lattice embedded on a genus two surface. Using numerical and analytical methods we show that the determinants of the Kasteleyn adjacency matrices have a dependence on the boundary conditions that, for large lattice size, can be expressed in terms of genus two theta functions. The period matrix characterizing the continuum limit of the lattice is computed using a discrete holomorphic structure. These results relate in a direct way the lattice combinatorics with conformal field theory, providing new insight to the lattice regularization of conformal field theories on higher genus Riemann surfaces.     

Critical Relaxation of a Three-Dimensional Fully Frustrated Ising Model

Critical relaxation from the low-temperature ordered state of the three-dimensional fully frustrated Ising model on a simple cubic lattice is studied by the short-time dynamics method. Cubic systems with periodic boundary conditions and linear sizes of L = 32, 64, 96, and 128 in each crystallographic direction are studied. Calculations were carried out by the Monte Carlo method using the standard Metropolis algorithm. The static critical exponents for the magnetization and correlation radius and the dynamic critical exponents are calculated.