Study of the three-dimensional Ising model on film geometry with the cluster Monte Carlo method

Topological properties of clusters are used to extract critical parameters. This method is tested for the bulk properties ofd=2 percolation and thed=2, 3 Ising model. For the latter we obtain an accurate value of the critical temperatureJ/k _B T _c=0.221617(18). In the case of thed=3 Ising model with film geometry the critical value of the surface coupling at the special transitions is determined as J_1c/J=1.5004(20) together with the critical exponents ₁ ^m =0.237(5) and=0.461(15).     

On the corrections to scaling in three-dimensional Ising models

The leading correction-to-scaling amplitudes for the spin-1/2, nearest-neighbor sc, bcc, and fee Ising models are considered with the particular aim of determining their signs. On the basis of previous two-variable series analyses by Chen, Fisher, and Nickel and renormalization group=4–d expansions, it is concluded that the correction amplitudes for the susceptibility, correlation length, specific heat, and spontaneous magnetization arenegative for all three lattices. Thus, for example, the effective exponent _eff(T) asymptotically approaches the true susceptibility exponent fromabove. Other earlier and more recent high-temperature series and field-theoretic analyses are seen to be consistent with this result. However, the usual nonasymptotic, perturbative field-theoretic approaches are essentially committed to positive correction amplitudes. The question of the signs therefore relates directly to the applicability of these non-asymptotic field-theoretic calculations to three-dimensional Ising models as well as to different experimental systems.     

On the critical behavior of the magnetization in high-dimensional Ising models

We derive rigorously general results on the critical behavior of the magnetization in Ising models, as a function of the temperature and the external field. For the nearest-neighbor models it is shown that ind4 dimensions the magnetization is continuous atT _c and its critical exponents take the classical values=3 and=1/2, with possible logarithmic corrections atd=4. The continuity, and other explicit bounds, formally extend tod>3 1/2. Other systems to which the results apply include long-range models ind=1 dimension, with 1/|x–y| couplings, for which 2/(–1) replacesd in the above summary. The results are obtained by means of differential inequalities derived here using the random current representation, which is discussed in detail for the case of a nonvanishing magnetic field.Research supported in part by NSF grant PHY-8301493 A02, and by a John S. Guggenheim Foundation fellowship (M.A.).     

Fast algorithm for the simulation of Ising models

Using a new microcanonical algorithm efficiently vectorized on a Cray XMP, we reach a simulation speed of 1.5 nsec per update of one spin, three times faster than the best previous method known to us. Data for the nonlinear relaxation with conserved energy are presented for the two-dimensional Ising model.     

Conformal invariance and surface defects in the two-dimensional Ising model. Exact results

The surface critical behavior of the two-dimensional Ising model with homogeneous perturbations in the surface interactions is studied on the one-dimensional quantum version. A transfer-matrix method leads to an eigenvalue equation for the excitation energies. The spectrum at the bulk critical point is obtained using anL ^–1 expansion, whereL is the length of the Ising chain. It exhibits the towerlike structure which is characteristic of conformal models in the case of irrelevant surface perturbations (h _s /J _s 0) as well as for the relevant perturbationh _s =0 for which the surface is ordered at the bulk critical point leading to an extraordinary surface transition. The exponents are deduced from the gap amplitudes and confirmed by exact finite-size scaling calculations. Both cases are finally related through a duality transformation.     

On the critical behavior of the Ising model with mixed two- and three-spin interactions

A study is made of a two-dimensional Ising model with staggered three-spin interactions in one direction and two-spin interactions in the other. The phase diagram of the model and its critical behavior are explored by conventional finite-size scaling and by exploiting relations between mass gap amplitudes and critical exponents predicted by conformal invariance. The model is found to exhibit a line of continuously varying critical exponents, which bifurcates into two Ising critical lines. This similarity of the model with the Ashkin-Teller model leads to a conjecture for the exact critical indices along the nonuniversal critical curve. Earlier contradictions about the universality class of the uniform (isotropic) case of the model are clarified.     

The distribution of clusters for the Ising model

We rigorously prove that the probabilityP _n for the origin to belong to a cluster of exactlyn positive spins in thev-dimensional Ising model behaves as exp(–n^{(v – 1)/v}) in various regions, including in particular the low-temperature positive and negative phases in zero magnetic field.     

Relaxation times in a finite Ising system with random impurities

Finite-size scaling effects of the Ising model with quenched random impurities are studied, focusing on critical dynamics. In contrast to the pure Ising model, disordered systems are characterized by continuous relaxation time spectra. Dynamic field theory is applied to compute the spectral densities of the magnetizationM(t) and ofM ²(t). In addition, universal cumulant ratios are calculated to second order in ^1/4, where =4–d andd<4 denotes the spatial dimension.     

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.     

Finite-size scaling properties of the damage distance and dynamical critical exponent for the Ising model

With the damage spreading method, scaling properties of the damage distance on the Ising model with heat bath dynamics are studied numerically. With the parallel flipping scheme, the scaling curves fall on two curves, which depend on the odd or even lattice sizes. The both scaling curves give the consistent dynamical exponent as z = 2.16±0.04 for d = 2 and z = 2.09±0.05 for d = 3, respectively. By shifting one of them, two curves overlap each other perfectly. Meanwhile, all the scaling curves obtained by single-spin flipping processes (with different odd or even lattice sizes) fall on a single curve, from which the consistent dynamical critical exponent with the parallel scheme is obtained z = 2.18±0.02 for d = 2 and z = 2.08±0.04 for d = 3.     

Critical dynamics of cluster algorithms in the dilute Ising model

Autocorrelation times for thermodynamic quantities atT _C are calculated from Monte Carlo simulations of the site-diluted simple cubic Ising model, using the Swendsen-Wang and Wolff cluster algorithms. Our results show that for these algorithms the autocorrelation timesdecrease when reducing the concentration of magnetic sites from 100% down to 40%. This is of crucial importance when estimating static properties of the model, since the variances of these estimators increase with autocorrelation time. The dynamical critical exponents are calculated for both algorithms, observing pronounced finite-size effects in the energy autocorrelation data for the algorithm of Wolff. We conclude that, when applied to the dilute Ising model, cluster algorithms become even more effective than local algorithms, for whichincreasing autocorrelation times are expected.     

Conformal invariance and the critical behavior of the tripletXY quantum chain

We introduce and study the critical properties of the tripletXY quantum chain. This system is described in terms of three-spin interactions and is the generalization of the standardXY quantum chain. We show that this model, with periodic boundaries, has a local gauge invariance and can be described by the composition of two triplet Ising models, with general toroidal boundary conditions. From this composition the phase diagram as well the conformai anomaly and critical exponents are determined by exploring their relations with the mass gap amplitudes predicted by conformai invariance.     

Essential singularity in percolation problems and asymptotic behavior of cluster size distribution

It is rigorously proved that the analog of the free energy for the bond and site percolation problem on in arbitrary dimension (> 1) has a singularity at zero external field as soon as percolation appears, whereas it is analytic for small concentrations. For large concentrations at least, it remains, however, infinitely differentiable and Borel-summable. Results on the asymptotic behavior of the cluster size distribution and its moments, and on the average surface-to-size ratio, are also obtained. Analogous results hold for the cluster generating function of any equilibrium state of a lattice model, including, for example, the Ising model, but infinite-range andn-body interactions are also allowed.Supported by the Fonds National Suisse de la Recherche Scientifique (to HK).     

Tree graph inequalities and critical behavior in percolation models

Various inequalities are derived and used for the study of the critical behavior in independent percolation models. In particular, we consider the critical exponent associated with the expected cluster sizex and the structure of then-site connection probabilities =_n(x₁,..., x_n). It is shown that quite generally 1. The upper critical dimension, above which attains the Bethe lattice value 1, is characterized both in terms of the geometry of incipient clusters and a diagramatic convergence condition. For homogeneousd-dimensional lattices with (x, y)=O(¦x -y¦^–(d–2+), atp=p _c, our criterion shows that =1 if > (6-d)/3. The connectivity functions _n are generally bounded by tree diagrams which involve the two-point function. We conjecture that above the critical dimension the asymptotic behavior of _n, in the critical regime, is actually given by such tree diagrams modified by a nonsingular vertex factor. Other results deal with the exponential decay of the cluster-size distribution and the function ₂ (x, y). A. P. Sloan Foundation Research Fellow. Research supported in part by the National Science Foundation Grant No. PHY-8301493.Research supported in part by the National Science Foundation Grant No. MCS80-19384.     

The magnetization-energy scaling limit in high dimension

In the single-phase region (including the critical point) of a nearest-neighbor Ising ferromagnet with zero external field, the block magnetization and energy within the infinite-volume system are, asymptotically for large block size, independent Gaussian variables when the dimensiond exceeds four. For other models, including ones with long-range interactions, a sufficient condition for such triviality of the scaling limit is finiteness of the bubble quantity.On leave from the University of Arizona, Tucson, Arizona 85721.     

Cluster algorithms for anisotropic quantum spin models

We present cluster Monte Carlo algorithms for theXYZ quantum spin models. In the special case ofS=1/2, the new algorithm can be viewed as a cluster algorithm for the 8-vertex model. As an example, we study theS=1/2XY model in two dimensions with a representation in which the quantization axis lies in the easy plane. We find that the numerical autocorrelation time for the cluster algorithm remains of the order of unity and does not show any significant dependence on the temperature, the system size, or the Trotter number. On the other hand, the autocorrelation time for the conventional algorithm strongly depends on these parameters and can be very large. The use of improved estimators for thermodynamic averages further enhances the efficiency of the new algorithms.