Interfacial free energy of the two-dimensional Ising model from the renormalization group

The interfacial free energy of a two-dimensional Ising model is calculated by using various renormalization group schemes. The results obtained are quantitatively consistent with known exact results. In addition, a general discussion of various drawbacks within different renormalization group approximations is given. The best result are obtained with the 4×4 finite cluster approximation, while the Migdal-Kadanoff approximation seems to be inherently unsuitable for calculation of interfacial properties.     

Domain growth in the three-dimensional dilute Ising model

We investigate the kinetics of domain growth in the three-dimensional Ising model with quenched random site dilution, using Monte Carlo simulation technique. A crossover from the power law growth regime to a much slower growth observed in our simulation is interpreted through the roughening of the interfaces by the quenched impurities. The results are also compared with the corresponding results in two dimensions.     

The shapes of bowed interfaces in the two-dimensional Ising model

We show that properly normalized net energy fluctuations associated with interfaces in two-dimensional Ising models are described, asymptotically, by random walk partition functions. Two examples are investigated: one is a droplet on a wall, and the other is two nearby, ideally parallel interfaces; the mean shapes of the interfaces in both cases prove to be elliptic, bowed outward from the wall or from each other, the semiminor axis of the latter ellipse being 1/2 that of the former, in accord with random walk results.     

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.     

The dynamic critical exponent of the three-dimensional Ising model

We measure the dynamic exponent of the three-dimensional Ising model using a damage spreading Monte Carlo approach as described by MacIsaac and Jan. We simulate systems fromL=5 toL=60 at the critical temperature,T _c =4.5115. We report a dynamic exponent,z=2.35±0.05, a value much larger than the consensus value of 2.02, whereas if we assume logarithmic corrections, we find thatz=2.05±0.05.     

Interface motion in a two-dimensional Ising model with a field

We determine by Monte Carlo simulations the width of an interface between the stable phase and the metastable phase in a two-dimensional Ising model with a magnetic field, in the case of nonconversed order parameter (Glauber dynamics). At zero temperature, the width increases ast with–1/3, as predicted by earlier theories. As temperature increases, the value of the effective exponent that we measure decreases toward the value 1/4, which is the value in the absence of magnetic field.     

Mathematical structure of the three-dimensional（3D） Ising model

An overview of the mathematical structure of the three-dimensional(3D) Ising model is given from the points of view of topology,algebra,and geometry.By analyzing the relationships among transfer matrices of the 3D Ising model,Reidemeister moves in the knot theory,Yang-Baxter and tetrahedron equations,the following facts are illustrated for the 3D Ising model.1) The complex quaternion basis constructed for the 3D Ising model naturally represents the rotation in a(3+1)-dimensional space-time as a relativistic quantum statistical mechanics model,which is consistent with the 4-fold integrand of the partition function obtained by taking the time average.2) A unitary transformation with a matrix that is a spin representation in 2 n·l·o-space corresponds to a rotation in 2n·l·o-space,which serves to smooth all the crossings in the transfer matrices and contributes the non-trivial topological part of the partition function of the 3D Ising model.3) A tetrahedron relationship would ensure the commutativity of the transfer matrices and the integrability of the 3D Ising model,and its existence is guaranteed by the Jordan algebra and the Jordan-von Neumann-Wigner procedures.4) The unitary transformation for smoothing the crossings in the transfer matrices changes the wave functions by complex phases φx,φy,and φz.The relationship with quantum field and gauge theories and the physical significance of the weight factors are discussed in detail.The conjectured exact solution is compared with numerical results,and the singularities at/near infinite temperature are inspected.The analyticity in β=1/(kBT) of both the hard-core and the Ising models has been proved only for β0,not for β=0.Thus the high-temperature series cannot serve as a standard for judging a putative exact solution of the 3D Ising model.     

Some numerical results on the block spin transformation for the 2D Ising model at the critical point

We study the block spin transformation for the 2D Ising model at the critical temperatureT _c . We consider the model with the constraint that the total spin in each block is zero. An old argument by Cassandro and Gallavotti strongly supports the Gibbsianness of the transformed measure, provided that such model has a critical temperatureT _c lower thanT _c . After describing a possible rigorous approach to the problem, we present numerical evidence that indeedT _c <T _c and study the Dobrushin-Shlosman uniqueness condition.     

Critical dynamics of the three-dimensional Ising model: A Monte Carlo study

Extensive Monte Carlo simulations have been performed to analyze the dynamical behavior of the three-dimensional Ising model with local dynamics. We have studied the equilibrium correlation functions and the power spectral densities of odd and even observables. The exponential relaxation times have been calculated in the asymptotic one-exponential time region. We find that the critical exponentz=2.09 ±0.02 characterizes the algebraic divergence with lattice size for all observables. The influence of scaling corrections has been analyzed. We have determined integrated relaxation times as well. Their dynamical exponentz _int agrees withz for correlations of the magnetization and its absolute value, but it is different for energy correlations. We have applied a scaling method to analyze the behavior of the correlation functions. This method verifies excellent scaling behavior and yields a dynamical exponentz _scal which perfectly agrees withz.     

Monte Carlo simulation of three-dimensional dilute Ising model

A multispin coding program for site-diluted Ising models on large simple cubic lattices is described in detail. The spontaneous magnetization is computed as a function of temperature, and the critical temperature as a function of concentration is found to agree well with the data of Marro et al.⁽⁴⁾ and Landau⁽³⁾ for smaller systems.The first successful epsilon expansion seems to be by D. E. Khmelnitskii,ZhETF 68:1960 (1975), English translationSov. Phys. JETP 41:981 (1975); for numerical estimates see K. E. Newman and E. K. Riedel,Phys. Rev. H25:264 (1982), for experiments see R. J. Birgenau, R. A. Cowley, G. Shirane and H. Yoshizawa,J. Stat. Phys. 34:817 (1984).     

采用正方元胞的Ising模型实空间重正化群解

在二维正方形晶格上,将元胞取为4格点正方形,采用3种不同的规则定义块自旋状态,进行了重正化群计算,得出了更为精确的结果;解决了元胞内格点数为偶数的重正化群计算问题.     

Interfaces of Ground States in Ising Models with Periodic Coefficients

We study the interfaces of ground states of ferromagnetic Ising models with external fields. We show that, if the coefficients of the interaction and the magnetic field are periodic, the magnetic field has zero flux over a period and is small enough, then for every plane, we can find a ground state whose interface lies at a bounded distance of the plane. This bound on the width of the interface can be chosen independent of the plane. We also study the average energy of the plane-like interfaces as a function of the direction. We show that there is a well defined thermodynamic limit for the energy of the interface and that it enjoys several convexity properties.     

Comment on a recent conjectured solution of the three-dimensional Ising model

In a recent paper published in Philosophical Magazine [Z.-D. Zhang, Phil. Mag. 87 (2007) p.5309], the author advances a conjectured solution for various properties of the three-dimensional Ising model. Here, we disprove the conjecture and point out the flaws in the arguments leading to the conjectured expressions.     

Complete wetting in the three-dimensional transverse Ising model

We consider a three-dimensional Ising model in a transverse magnetic fieldh and a bulk fieldH. An interface is introduced by an appropriate choice of boundary conditions. At the point (H=0,h=0) spin configurations corresponding to different positions of the interface are degenerate. By studying the phase diagram near this multiphase point using quantum mechanical perturbation theory, we show that the quantum fluctuations, controlled byh, split the multiphase degeneracy giving rise to an infinite sequence of layering transitions.     

Ferroelectricity in a diatomic Ising chain as investigated by the elastic Ising model

An elastic Ising model for a one-dimensional diatomic spin chain is proposed to explain the ferroelectricity induced by the collinear magnetic order with a low-excited energy state. A statistical theory based on this model is developed to calculate the electrical and magnetic properties of Ca₃CoMnO₆, a typical quasi-one-dimensional diatomic spin chain system. The calculated ferroelectric polarization and dielectric susceptibility show a good agreement with recently reported data on Ca₃Co_2-xMn_xO₆ (x ≈0.96) (Phys. Rev. Lett. 100 047601 (2008)), although the predicted magnetic susceptibility does not coincide well with experiment. We also address the rationality and deficiency of this model by including a first-order correction which improves the consistency between the model and experiment.     

Renormalization Group Study of Ising Transition in Double-Frequency sine-Gordon Model

We utilize the renormalization group (RG) technique to analyze the Ising critical behavior in the doublefrequency sine-Gordon model. The one-loop RG equations obtained show unambiguously that there exist two Isingcritical points besides the trivial Gaussian fixed point. The topology of the RG flows is obtained as well.     

Ferroelectricity in diatomic Ising chain as investigated by elastic Ising model

An elastic Ising model for a one-dimensional diatomic spin chain is proposed to explain the ferroelectricity induced by the collinear magnetic order with a low-excited energy state. A statistical theory based on this model is developed to calculate the electrical and magnetic properties of Ca₃CoMnO₆, a typical quasi-one-dimensional diatomic spin chain system. The calculated ferroelectric polarization and dielectric susceptibility show a good agreement with recently reported data on Ca₃Co_2-xMn_xO₆ (x ≈0.96) (Phys. Rev. Lett. 100 047601 (2008)), although the predicted magnetic susceptibility does not coincide well with experiment. We also address the rationality and deficiency of this model by including a first-order correction which improves the consistency between the model and experiment.     

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.     

Metastability and the Ising model

We discuss a recent theorem which establishes a precise connection between (i) the approximate degeneracy of the zero eigenvalue for the generator of the Glauber dynamics of the Ising model in a small nonzero field and below the critical temperature, (ii) the existence of a partition of the configuration space into a normal region and a metastable region. This enables us to demonstrate that the recent approach to metastability of Davies and Martin may be viewed as a simple (although in some ways fairly crude) approximation to the conventional approach. We also obtain what appear to be the first results concerning the stability of metastable states under small perturbations.