Susceptibility of the rectangular Ising ferromagnet

The critical index values= 7/4 for the susceptibility and=15 for the critical isotherm are derived rigorously for the rectangular Ising ferromagnet with nearest neighbor interactions. The critical indices associated with the Fisher moment definition of the correlation length are obtained asTT _c+. The index of the fluctuation sum definition of critical correlations is obtained.Partially supported by grant PHY 76 17191.     

Isotropic majority-vote model on a square lattice

The stationary critical properties of the isotropic majority vote model on a square lattice are calculated by Monte Carlo simulations and finite size analysis. The critical exponents, , and are found to be the same as those of the Ising model and the critical noise parameter is found to beq _c =0.075±0.001.     

Two-Dimensional Dilute Ising Models: Critical Behavior near Defect Lines

We consider two-dimensional Ising models with randomly distributed ferromagnetic bonds and study the local critical behavior at defect lines by extensive Monte Carlo simulations. Both for ladder- and chain-type defects, nonuniversal critical behavior is observed: the critical exponent of the defect magnetization is found to be a continuous function of the strength of the defect coupling. Analyzing corresponding stability conditions, we obtain new evidence that the critical exponent of the bulk correlation length of the random Ising model does not depend on dilution, i.e., =1.     

The Hausdorff dimension of random walks and the correlation length critical exponent in Euclidean field theory

We study the random walk representation of the two-point function in statistical mechanics models near the critical point. Using standard scaling arguments, we show that the critical exponentv describing the vanishing of the physical mass at the critical point is equal tov /d_w, whered _w is the Hausdorff dimension of the walk, andv is the exponent describing the vanishing of the energy per unit length of the walk at the critical point. For the case ofO(N) models, we show thatv ₀=, where is the crossover exponent known in the context of field theory. This implies that the Hausdorff dimension of the walk is/v forO(N) models.     

Scaling of particle trajectories on a lattice

The scaling behavior of the closed trajectories of a moving particle generated by randomly placed rotators or mirrors on a square or triangular lattice is studied numerically. On both lattices, for most concentrations of the scatterers the trajectories close exponentially fast. For special critical concentrations infinitely extended trajectories can occur which exhibit a scaling behavior similar to that of the perimeters of percolation clusters.At criticality, in addition to the two critical exponents =15/7 andd _f=7/4 found before, the critical exponent =3/7 appears. This exponent determines structural scaling properties of closed trajectories of finite size when they approach infinity. New scaling behavior was found for the square lattice partially occupied by rotators, indicating a different universality class than that of percolation clusters.Near criticality, in the critical region, two scaling functions were determined numerically:f(x), related to the trajectory length (S) distributionn _s, andh(x), related to the trajectory sizeR _s (gyration radius) distribution, respectively. The scaling functionf(x) is in most cases found to be a symmetric double Gaussian with the same characteristic size exponent =0.433/7 as at criticality, leading to a stretched exponential dependence ofn _S onS, n_Sexp(–S ^6/7). However, for the rotator model on the partially occupied square lattice an alternative scaling function is found, leading to a new exponent =1.6±0.3 and a superexponential dependence ofn _S onS.h(x) is essentially a constant, which depends on the type of lattice and the concentration of the scatterers. The appearance of the same exponent =3/7 at and near a critical point is discussed.     

Statistical mechanics of polymer networks of any topology

The statistical mechanics is considered of any polymer network with a prescribed topology, in dimensiond, which was introduced previously. The basic direct renormalization theory of the associated continuum model is established. It has a very simple multiplicative structure in terms of the partition functions of the star polymers constituting the vertices of the network. A calculation is made toO(²), whered=4–, of the basic critical dimensions _L associated with anyL-leg vertex (L1). From this infinite series of critical exponents, any topology-dependent critical exponent can be derived. This is applied to the configuration exponent _G of any networkG toO(²), includingL-leg star polymers. The infinite sets of contact critical exponents between multiple points of polymers or between the cores of several star polymers are also deduced. As a particular case, the three exponents ₀, ₁, ₂ calculated by des Cloizeaux by field-theoretic methods are recovered. The limiting exact logarithmic laws are derived at the upper critical dimensiond=4. The results are generalized to the series of topological exponents of polymer networks near a surface and of tricritical polymers at the-point. Intersection properties of networks of random walks can be studied similarly. The above factorization theory of the partition function of any polymer network over its constitutingL-vertices also applies to two dimensions, where it can be related to conformal invariance. The basic critical exponents _L and thus any topological polymer exponents are then exactly known. Principal results published elsewhere are recalled.     

Effective two-dimensional description from critical phenomena in black holes

We study the occurrence of critical phenomena in four-dimensional, rotating and charged black holes, derive the critical exponents and show that they fulfill the scaling laws. Correlation function critical exponents and Renormalization Group considerations assign an effective (spatial) dimension,d=2, to the system. The two-dimensional Gaussian approximation to critical systems is shown to reproduce all the black hole's critical exponents. Higher order corrections (which are always relevant) are discussed. Identifying the two-dimensional surface with the event horizon and noting that generalization of scaling leads to conformal invariance and then to string theory, we arrive at 't Hooft's string interpretation of black holes. From this, a model for dealing with a coarse grained black hole quantization is proposed. We also give simple arguments that lead to a rough quantization of the black hole mass in units of the Planck mass, i.e.M(1/2)M _Pll with anl positive integer and then, from this result, to the proportionality between quantum entropy and area.This essay received the fifth award from the Gravity Research Foundation, 1994—Ed.     

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).     

New approaches inμ + SR studies on critical phenomena in Ni

The critical phenomena in Ni are probed by pulsedSR method under longitudinal- and zero external magnetic fields. The sample magnetization around the critical temperature is confirmed simultaneously by bulk magnetization measurement in situ, disappearance of transverseSR signal and recovery of asymmetry under longitudinal field. At the same time, the ratio of the ⁺ hyperfine field to the bulk magnetization in the ferromagnetic phase below the critical temperature is determined from the observables obtained only in the present experiment. The zero- and low-field longitudinal relaxation rate of muon does not diverge in approaching toT _c in the paramagnetic region, but seems to reach a saturation value.This work is supported by the Grand-in-Aid of the Japanese Ministry of Education, Culture and Science.     

The Scaling Limit of the Incipient Infinite Cluster in High-Dimensional Percolation. I. Critical Exponents

This is the first of two papers on the critical behavior of bond percolation models in high dimensions. In this paper, we obtain strong joint control of the critical exponents and for the nearest neighbor model in very high dimensions d6 and for sufficiently spread-out models in all dimensions d>6. The exponent describes the low-frequency behavior of the Fourier transform of the critical two-point connectivity function, while describes the behavior of the magnetization at the critical point. Our main result is an asymptotic relation showing that, in a joint sense, =0 and =2. The proof uses a major extension of our earlier expansion method for percolation. This result provides evidence that the scaling limit of the incipient infinite cluster is the random probability measure on ^d known as integrated super-Brownian excursion (ISE), in dimensions above 6. In the sequel to this paper, we extend our methods to prove that the scaling limits of the incipient infinite cluster's two-point and three-point functions are those of ISE for the nearest neighbor model in dimensions d6.     

Mean field theory of self-organized critical phenomena

A mean field theory is presented for the recently discovered self-organized critical phenomena. The critical exponents are calculated and found to be the same as the mean field values for percolation. The power spectrum has 1/f behavior with exponentg4=1.     

Correlations in inhomogeneous Ising models

Using general methods developed in a previous treatment we study correlations in inhomogeneous Ising models on a square lattice. The nearest neighbour couplings can vary both in strength and sign such that the coupling distribution is translationally invariant in horizontal direction. We calculate correlations parallel to the layering in the horizontally layered model with periodv=2. If the model has a finite critical temperature,T _c>0, the order parameter in the frustrated case may become discontinuous forT0. Correlations atT=T _c decay algebraically with critical exponent =1/4 and exponentially forT>T _c. If the critical temperature vanishes,T _c=0, we always have exponential decay at finite temperatures, while atT=T _c=0 we encounter either long-range order or algebraic decay with critical index =1/2, i.e.T=0 is thus a critical point.Work performed within the research program of the Sonder forschungsbereich 125 Aachen-Jülich-Köln     

Existence of the critical point in φ4 field theory

We consider the ⁴ quantum field theory in two and three spacetime dimensions. In the single phase region the physical mass (inverse correlation length)m() decreases continuously to zero as the bare mass parameter approaches a critical value _c from above. In three dimensions the critical point _c is in the single phase region and the physical mass vanishes there,m( _c )=0.A consequence of our results is that the critical exponentv governing the approach to infinite correlations is bounded below (rigorously) by its classical value, 1/2.Supported in part by the National Science Foundation under Grant MPS74-13252     

Spectrum and eigenfunctions of the Frobenius-Perron operator of the tent map

The spectrum and eigenfunctions of the Frobenius-Perron operator induced by the tent map are discussed in detail. Special attention is paid to the case where the critical point of the map lies on an aperiodic trajectory and the differences from maps with a periodic critical trajectory are stressed. It is shown that the relevant eigenvalues of the spectrum are not sensitive to the aperiodicity of the critical trajectory. All other parts of the spectrum and all eigenfunctions in particular are changed drastically if the critical trajectory becomes aperiodic. The intimate connection between the point spectrum and the kneading invariant is established and the critical slowing down as well as the band splitting are a consequence of its properties. The structure of the infinite sequence of null spaces and its implications on the spectrum of the operator are discussed. It is shown that any initial distributionP(0,x) of bounded variation can be projected uniquely onto the eigenfunctions of the relevant eigenvalues and that the time dependence ofP(n, x) is determined by this expansion up to an errorO( ⁿ). From this the stationary and the asymptotic behavior of the correlation function x(n) x can be derived exactly.     

Note on the quantum displacement of the critical point

The quantum corrections to the law of corresponding states are studied by calculating the critical pressure, temperature, and density to first order in Planck's constanth on an exactly soluble model. The ratio of the critical parameters to the corresponding classical values are found to be (p _c/p _c ⁰)^1/2=_c/_c ⁰ = T_c/T_c ⁰ = 1–0.67, with=h _c ^1/3(mkT _c)^–1/2. The critical ratio is independent ofh to first order. The results are compared with critical data for noble gases and hydrogen isotopes.     

Fractal dimension and grand Universality of critical phenomena

Conformation of branched random fractals formed in equilibrium processes is discussed using a Flory-type theory. Within this approach we find only three distinct types or classes of random fractals. We call these theextended, thecompensated, and thecollapsed states. In particular, the critical clusters in thermal phase transitions are found to be of the compensated type and have approximately the same value of the fractal dimension. The Flory theory predicts the upper critical dimension for these clusters to be 6 instead of 4. This result and the apparent grand universality of the fractal geometry of the clusters in critical phenomena are discussed.     

Correlation length and its critical exponent for percolation processes

Some critical exponent inequalities are given involving the correlation length of site percolation processes on ^d. In particular, it is shown thatv2/d, which implies that the critical exponentv cannot take its mean-field value for the three-dimensional percolation processes.     

On the equivalence of different order parameters and coexistence of phases for Ising ferromagnet. II

We investigate Ising spin systems with general ferromagnetic, translation invariant interactions,H=–J _BB,J _B0. We show that the critical temperatureT _i for the order parameterp _i defined as the temperature below whichp _i>0, is independent of the way in which the symmetry breaking interactions approach zero from above. Furthermore, all the equivalent correlation functions have the same critical exponents asT T_i from below, e.g. for pair interactions all the odd correlations have the same critical index as the spontaneous magnetization. The number of fluid and crystalline phases (periodic equilibrium states) coexisting at a temperatureT at which the energy is continuous is shown to be related to the number of symmetries of the interactions. This generalizes previous results for Ising spins with even (and non-vanishing nearest-neighbour) ferromagnetic interactions. We discuss some applications of these results to the triangular lattice with three body interactions and to the Ashkin-Teller model. Our results give the answer to the question raised by R.J. Baxter et al. concerning the equality of some critical exponents.Supported by NSF Grant PHY 77-22302     

Renormalization group treatment of the quantum nonlinear σ-model

The quantum nonlinear -model in (d+1)-dimensional space-time is investigated by a renormalization group approach. The beta-functions for the couplingg and the temperaturet are given. The renormalisation group equations of theN-point functions are derived for finite coupling and finite temperature. It is known that the model shows a phase transition at zero temperature at some critical couplingg _c . The behaviour near this critical point is investigated. The crossover exponent , describing the crossover between different regimes near the critical point is calculated, verifying a conjecture by Chakravarty, Halperin and Nelson, who have argued that ind dimensions should have the same value as the critical exponent of the correlation length in a (d+1)-dimensional classical system. A subtraction scheme appropriate to calculate the renormalisation factors and from these the beta-functions at finite temperature and finite coupling constant will be introduced. Using this method the beta-functions will be calculated to order two loops. The exponents obtained this way are in good agreement with the values found on other ways.     

Mean-Field Theory for Percolation Models of the Ising Type

The q=2 random cluster model is studied in the context of two mean-field models: the Bethe lattice and the complete graph. For these systems, the critical exponents that are defined in terms of finite clusters have some anomalous values as the critical point is approached from the high-density side, which vindicates the results of earlier studies. In particular, the exponent ~ which characterizes the divergence of the average size of finite clusters is 1/2, and ~, the exponent associated with the length scale of finite clusters, is 1/4. The full collection of exponents indicates an upper critical dimension of 6. The standard mean field exponents of the Ising system are also present in this model (=1/2, =1), which implies, in particular, the presence of two diverging length-scales. Furthermore, the finite cluster exponents are stable to the addition of disorder, which, near the upper critical dimension, may have interesting implications concerning the generality of the disordered system/correlation length bounds.