Spiral Solutions of the Two-Dimensional Complex Ginzburg-Landau Equation

The multi-order exact solutions of the two-dimensional complex Ginzburg-Landau equation are obtained by making use of the wave-packet theory. In these solutions, the zeroth-order exact solution is a plane wave, the first-order exact solutions are shock waves for the amplitude and spiral waves both between the amplitude and the shift of phase and between the shift of phase and the distance.     

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.     

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 Transitions in the Two-Dimensional Slightly Diluted Five-Vertex Potts Model

Physics of the Solid State - Phase transitions in the two-dimensional slightly diluted Potts model on a square lattice at q = 5 have been studied using the computer simulation method. The systems...     

Efficient Algorithms for the Two-Dimensional Ising Model with a Surface Field

The bond propagation and site propagation algorithms are extended to the two-dimensional (2D) Ising model with a surface field. With these algorithms we can calculate the free energy, internal energy, specific heat, magnetization, correlation functions, surface magnetization, surface susceptibility and surface correlations. The method can handle continuous and discrete bond and surface-field disorder and is especially efficient in the case of bond or site dilution. To test these algorithms, we study the wetting transition of the 2D Ising model, which was solved exactly by Abraham. We can locate the transition point accurately with a relative error of \(10^{-8}\) . We carry out the calculation of the specific heat and surface susceptibility on lattices with sizes up to \(200^2 \times 200\) . The results show that a finite jump develops in the specific heat and surface susceptibility at the transition point as the lattice size increases. For lattice size \(320^2 \times 320\) the parallel correlation length exponent is \(1.86\) , while Abraham’s exact result is \(2.0\) . The perpendicular correlation length exponent for lattice size \(160^2\times 160\) is \(1.05\) , whereas its exact value is \(1.0\) .     

On the Singularities in the Susceptibility Expansion for the Two-Dimensional Ising Model

For temperatures below the critical temperature, the magnetic susceptibility for the two-dimensional isotropic Ising model can be expressed in terms of an infinite series of multiple integrals. With respect to a parameter related to temperature and the interaction constant, the integrals may be extended to functions analytic outside the unit circle. In a groundbreaking paper, Nickel (J Phys A 32:3889–3906, 1999) identified a class of singularities of these integrals on the unit circle. In this note we show that there are no other singularities on the unit circle.     

Critical Exponents of the Diluted Ising Model between Dimensions 2 and 4

Within the massive field-theoretic renormalization-group approach the expressions for the and functions of the anisotropic mn-vector model are obtained for general space dimension d in three-loop approximation. Resumming corresponding asymptotic series, critical exponents for the case of the weakly diluted quenched Ising model (m = 1, n = 0), as well as estimates for the marginal order parameter component number m _c of the weakly diluted quenched m-vector model, are calculated as functions of d in the region 2 d < 4. Conclusions concerning the effectiveness of different resummation techniques are drawn.     

Analytic Verification of the Droplet Picture in the Two-Dimensional Ising Model

It is widely accepted that the free energy F(H) of the two-dimensional Ising model in the ferromagnetic phase T<T _c has an essential branch cut singularity at the origin H=0. The phenomenological droplet theory predicts that near the cut drawn along the negative real axis H<0, the imaginary part of the free energy per lattice site has the form ImF[exp(±i)|H|]=±B|H|exp(–A/|H|) for small |H|. We verify this prediction in analytical perturbative transfer matrix calculations for the square lattice Ising model for all temperatures 0<T<T _c and arbitrary anisotropy ratio J ₁/J ₂. We obtain an expression for the constant A which coincides exactly with the prediction of the droplet theory. For the amplitude B we obtain B=M/18, where M is the equilibrium spontaneous magnetization. In addition we find discrete-lattice corrections to the above mentioned phenomenological formula for ImF, which oscillate in H ^–1.     

Metastability in the Two-Dimensional Ising Model with Free Boundary Conditions

We investigate metastability in the two dimensional Ising model in a square with free boundary conditions at low temperatures. Starting with all spins down in a small positive magnetic field, we show that the exit from this metastable phase occurs via the nucleation of a critical droplet in one of the four corners of the system. We compute the lifetime of the metastable phase analytically in the limit T 0, h 0 and via Monte Carlo simulations at fixed values of T and h and find good agreement. This system models the effects of boundary domains in magnetic storage systems exiting from a metastable phase when a small external field is applied.     

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 Direct Calculation of Critical Exponents of Two-Dimensional Anisotropic Ising Model

Using an exact solution of the one-dimensional quantum transverse-field Ising model, we calculate the critical exponents of the two-dimensional anisotropic classical Ising model (IM). We verify that the exponents are the same as those of isotropic classical IM. Our approach provides an alternative means of obtaining and verifying these well-known results.     

Universal Amplitude Ratios in the Critical Two-Dimensional Ising Model on a Torus

Using results from conformal field theory, we compute several universal amplitude ratios for the two-dimensional Ising model at criticality on a symmetric torus. These include the correlation-length ratio x =lim _L (L)/L and the first four magnetization moment ratios V _2n = ²ⁿ / ^{2 n} . As a corollary we get the first four renormalized 2n-point coupling constants for the massless theory on a symmetric torus, G*_2n . We confirm these predictions by a high-precision Monte Carlo simulation.     

Thermodynamic and Magnetic Properties of the Two-Dimensional Anisotropic Ising Model with Competing Interactions

Physics of the Solid State - The two-dimensional anisotropic Ising model with competing interactions is studied on a square lattice by Monte Carlo methods using the Wang–Landau algorithm. The...     

Stationary State Solutions of a Bond Diluted Kinetic Ising Model: An Effective-Field Theory Analysis

We examined the stationary state solutions of a bond diluted kinetic Ising model under a time dependent oscillating magnetic field within the effective-field theory (EFT) for a honeycomb lattice (q=3). The effects of the Hamiltonian parameters on the dynamic phase diagrams have been discussed in detail. Bond dilution process on the kinetic Ising model causes a number of interesting and unusual phenomena such as reentrant phenomena and has a tendency to destruct the first-order transitions and the dynamic tricritical point. Moreover, we have investigated the variation of the bond percolation threshold as functions of the amplitude and frequency of the oscillating field.     

随机Ginzburg-Landau方程的数值模拟

对随机Ginzburg-Landau方程进行数值研究,构造一个非线性差分格式和一个线性化差分格式.通过对确定性和随机Ginzburg-Landau方程的计算,表明所构造的格式具有较高的精度和较快的计算效率.对随机Ginzburg-Landau方程就噪声振幅的不同取值进行了数值模拟,并对由此引发的各种行为进行了描述.     

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).     

Diluted transverse Ising model within the mean field renormalization group

Clusters of different size and symmetry are exploited in the study of the diluted transverse Ising model on several lattices within the mean field renormalization group approach. It is noticed that the critical exponents depend both on the size of clusters as well as on the cluster symmetry. Harris' conjecture is verified for all lattices studied.