Upper bounds onT c for one-dimensional Ising systems

We present upper bounds on the critical temperature of one-dimensional Ising models with long-range,l/n interactions, where 1<2. In particular for the often studied case of =2 we have an upper bound onT _c which is less than theT _c found by a number of approximation techniques. Also for the case where is small, such as =1.1, we obtain rigorous bounds which are extremely close, within 1.0%, to those found by approximation methods.     

Fluctuations and lack of self-averaging in the kinetics of domain growth

The fluctuations occurring when an initially disordered system is quenched at timet=0 to a state, where in equilibrium it is ordered, are studied with a scaling theory. Both the mean-sizel(t) ^d of thed-dimensional ordered domains and their fluctuations in size are found to increase with the same power of the time; their relative size fluctuations are independent of the total volumeL ^d of the system. This lack of self-averaging is tested for both the Ising model and the ⁴ model on the square lattice. Both models exhibit the same lawl(t)=(Rt) ^x withx=1/2, although the ⁴ model has soft walls. However, spurious results withx1/2 are obtained if bad pseudorandom numbers are used, and if the numbern of independent runs is too small (n itself should be of the order of 10³). We also predict a critical singularity of the rateR(1–T/T _c) ^v(z–1/x),v being the correlation length exponent,z the dynamic exponent.Also quenches to the critical temperatureT _c itself are considered, and a related lack of self-averaging in equilibrium computer simulations is pointed out for quantities sampled from thermodynamic fluctuation relations.     

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 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.     

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.     

Higher-order susceptibilities of the regular and the random Ising model on the Cayley tree. I

Explicit expressions for the fourth-order susceptibility (⁴), the fourth derivative of thebulk free energy with respect to the external field, are given for the regular and the random-bond Ising model on the Cayley tree in the thermodynamic limit, at zero external field. The fourth-order susceptibility for the regular system diverges at temperature T _c ⁽⁴⁾ = 2k _B ^–1 J/ln{1+2/[(z–1)^3/4–1]}, confirming a result obtained by Müller-Hartmann and Zittartz [Phys. Rev. Lett. 33:893 (1974)]; Herez is the coordination number of the lattice,J is the exchange integral, andk _B is the Boltzmann constant. The temperatures at which ⁽⁴⁾ and the ordinary susceptibility (²) diverge are given also for the random-bond and the random-site Ising model and for diluted Ising models.     

Flux diffusion in high-T c superconductors

In type-II superconductors in the flux flow (J J _c ), flux creep (J _c J _c ), and thermally activated flux flow (TAFF) (J J _c ) regimes the inductionB(r,t), averaged over several penetration depths , in general follows from a nonlinear equation of motion into which enter the nonlinear resistivities (B, J ,T) caused by flux motion and (B, J ,T) caused by other dissipative processes.J andJ are the current densities perpendicular and parallel toB,B=|B|, andT is the temperature. For flux flow and TAFF in isotropic superconductors with weak relative spatial variation ofB, this equation reduces to the diffusion equation plus a correction term which vanishes whenJ =0 (this means B××B=0) or when – = 0 (isotropic normal conductor). When this diffusion equation holds the material anisotropy may be accounted for by a tensorial . The response of a superconductor to an applied current or to a change of the applied magnetic field is considered for various geometries. Such perturbations affect only a surface layer of thickness where a shielding current flows which pulls at the flux lines; the resulting deformation of the vortex lattice diffuses into the interior until a new equilibrium or a new stationary state is reached. The a.c. response, in particular the frequency with maximum damping, depends thus on the geometry and size of the superconductor.     

On scaling properties of cluster distributions in Ising models

Scaling relations of cluster distributions for the Wolff algorithm are derived. We found them to be well satisfied for the Ising model ind=3 dimensions. Using scaling and a parametrization of the cluster distribution, we determine the critical exponent/=0.516(6) with moderate effort in computing time.     

T c =0-Ordering in the ruby Ising-spin glass?

Isothermal and adiabatic susceptibilities characterize the rubies under investigation, (Cr _x Al_1–x )₂O₃ with 0.00025x0.008, as symmetric Ising spin glasses with a rms interaction of . Spin-freezing phenomena, like cusps, plateaus, and thermoremanence in the low-field magnetization are associated with a restricted heat transfer between the spin system and the thermal bath in the precritical region (TJ). At equilibrium andTJ, the scaling of the non-linear magnetization and the slowing down of the zerofield, average relaxation rate indicate a spin glass transition atT _c =0. Among possible reasons for this apparent discrepancy to theT _c J-hypothesis by Ogielski et al., are internal random fields.     

A numerical study of the hierarchical Ising model: High-temperature versus epsilon expansion

We study numerically the magnetic susceptibility of the hierarchical model with Ising spins (=±1) above the critical temperature and for two values of the epsilon parameter. The integrations are performed exactly using recursive methods which exploit the symmetries of the model. Lattices with up to 2¹⁸ sites have been used. Surprisingly, the numerical data can be fitted very well with a simple power law of the form (1-/ ₀)^g for thewhole temperature range considered. This approximate law implies a simple approximate formula for the coefficients of the high-temperature expansion, and, more importantly, approximate relations among the coefficients themselves. We found that some of these approximate relations hold with errors less then 2%. On the other hand,g differs significantly from the critical exponent calculated with the epsilon expansion, even when the fit is restricted to intervals closer to ^c. We discuss this discrepancy in the context the renormalization group analysis of the hierarchical model.     

Critical properties and finite-size effects of the five-dimensional Ising model

Monte Carlo calculations of the thermodynamic properties (energy, specific heat, magnetization suceptibility, renormalized coupling) of the nearest-neighbour Ising ferromagnet on a five-dimensional hypercubic lattice are presented and analyzed. Lattices of linear dimensionsL=3, 4, 5, 6, 7 with periodic boundary conditions are studied, and a finite size scaling analysis is performed, further confirming the recent suggestion thatL does not scale with the correlation length (the temperature variation of which near the critical temperatureT _c is |1-T/T _c |^–1/2), but rather with a thermodynamic lengthl (withl|1-T/T _c |^–2/d ,d=5 here). The susceptibility (extrapolated to the thermodynamic limit) agrees quantitatively with high temperature series extrapolations of Guttmann. The problem of fluctuation corrections to the leading (Landau-like) critical behaviour is briefly discussed, and evidence given for a specific-heat singularity of the form |1-T/T _c |^1/2, superimposed on its leading jump.Dedicated to Prof. Dr. H.E. Müser on the occasion of his 60th birthday     

Discontinuity of the magnetization in one-dimensional 1/¦x−y¦2 Ising and Potts models

Results from percolation theory are used to study phase transitions in one-dimensional Ising andq-state Potts models with couplings of the asymptotic formJ _x,y const/¦x–y¦². For translation-invariant systems with well-defined lim _x x ² J _x =J ⁺ (possibly 0 or ) we establish: (1) There is no long-range order at inverse temperatures withJ ⁺1. (2) IfJ ⁺>q, then by sufficiently increasingJ ₁ the spontaneous magnetizationM is made positive. (3) In models with 0<J ⁺< the magnetization is discontinuous at the transition point (as originally predicted by Thouless), and obeysM( _c )1/( _c J ⁺)^1/2. (4) For Ising (q=2) models withJ ⁺<, it is noted that the correlation function decays as xy()c()/|x–y|² whenever< _c . Points 1–3 are deduced from previous percolation results by utilizing the Fortuin-Kasteleyn representation, which also yields other results of independent interest relating Potts models with different values ofq.     

Critical temperature for two-dimensional ising ferromagnets with quenched bond disorder

The configuration-averaged free energy of a quenched, random bond Ising model on a square lattice which contains an equal mixture of two types of ferromagnetic bonds J₁ and J₂ is shown to obey the same duality relation as the ordered rectangular model with the same two bond strengths. If the random.system has a single, sharp critical point, the critical temperature T_c must be identical to that of the ordered system, i.e., sinh(2J ₁/kT _c) sinh(2J ₂/kT _c) = 1. Since _c ^(B) = 1/2, we can takeJ ₂ 0 and use Bergstresser-type inequalities to obtain(/dp) exp(–2J ₁/kT_c¦_p=p_c + = 1, in agreement with Bergstresser's rigorous result for the diluted ferromagnet near the percolation threshold.Work supported in part by National Science Foundation Grant No. DMR 76-21703, Office of Naval Research Grant No. N00014-76-C-0106, and National Science Foundation MRL program Grant No. DMR 76-00678.Paper presented at the 37th Yeshiva University Statistical Mechanics Meeting, May 10, 1977.     

Corrections to the critical temperature in 2D Ising systems with Kac potentials

We consider ad=2 Ising system with a Kac potential whose mean-field critical temperature is 1. Calling >0 the Kac parameter, we prove that there existsc ^*>0 so that the true inverse critical temperature _cr() > 1 +by ² log ^-1, for anyb<c ^* and correspondingly small. We also show that if 0 andbc ^*, suitably, then the correlation functions (normalized and rescaled) converge to those of a non-Gaussian Euclidean field theory.     

On the critical exponent for random walk intersections

The exponent _d for the probability of nonintersection of two random walks starting at the same point is considered. It is proved that 1/2<₂3/4. Monte Carlo simulations are done to suggest ₂=0.61 and ₃0.29.     

High-resolution laser spectroscopy of theQ v(0) transitions in solid parahydrogen

Our recent high-resolution laser spectroscopy of theQ _v(0) (=0,J=00) transitions in solid parahydrogen is discussed. The systems studied include the fundamental vibrational bands of impurity D₂ and HD, the first and second overtones of parahydrogen, and the charge-induced spectrum of-ray irradiated parahydrogen. Additionally, Stark and stimulated Raman-gain spectroscopies are applied to the solid. The linewidths are as sharp as 2 MHz HWHM, which is highly unusual for a solid. Our spectra demonstrate a variety of physical phenomena, particularly thek = 0 selection rule, as well asJ = 1/J = 0 pair intermolecular interactions.     

Effective-field theory of spin glasses and the coherent-anomaly method. I

A new cluster-effective-field theory of spin glasses is formulated. Basic formulas for the spin-glass transition point and the spin-glass susceptibility in the high-temperature phase are obtained. The present theory combined with the coherent-anomaly method is shown to be useful to estimate the true critical point and the nonclassical critical exponent of a spin-glass transition. Concerning the two-dimensional ±J model, we have _s =5.2(1) forT _SG=0, which agrees well with the data by some other authors. As for the threedimensional±J model, the present tentative analysis givesT _SG=1.2(1)(J/k _B) and _s =4(1), but more extensive calculations are needed.     

High-temperature series for scalar-field Lattice models: Generation and analysis

An implementation of the free-embedding scheme for high-temperature series generation on the body-centered cubic family of lattices in arbitrary dimensiond is, described. Series to order 21 in inverse temperature are tabulated for several scalar field models, both for the magnetic susceptibility and for the second moment of the spin correlation function. The critical behavior of a family of 3-dimensional double Gaussian models, which interpolate continuously between the spin-1/2 Ising model and the Gaussian model, is analyzed in detail away from the Gaussian model limit using confluent inhomogeneous secondorder differential approximants. With our best estimate of the correction-to-scaling exponent,=0.52±0.03, the leading exponents for the susceptibility and correlation length for this family are consistent with universality and are given by=1.237±0.002 and =0.630±0.0015, respectively, and=2–/=0.0359±0.0007.     

Critical behavior of short range Potts glasses

We study by means of Monte Carlo simulations and the numerical transfer matrix technique the critical behavior of the short rangep=3 state Potts glass model in dimensionsd=2,3,4 with both Gaussian and bimodal (±J) nearest neighbor interactions on hypercubic lattices employing finite size scaling ideas. Ind=2 in addition the degeneracy of the glass ground state is computed as a function of the number of Potts states forp=3, 4, 5 and compared to that of the antiferromagnetic ground state. Our data indicate a transition into a glass phase atT=0 ind=2 with an algebraic singularity, aT=0 transition ind=3 with an essential singularity of the form exp(const.T ^–2), and an algebraic singularity atT0.25 ind=4. We conclude that the lower critical dimension of the present model isd _c =3 or very close to it. Some of the critical exponents are estimated and their respective values discussed.     

Critical dynamics of finite Ising model

The Glauber kinetics of Ising spins is considered as a queueing process and simulated event by event as first proposed by Bortz, Kalos, and Lebowitz. The advantage of this algorithm compared to the standard single-flip Monte Carlo method is discussed for the situation of slowing down of dynamics. This process is used to generate fluctuations of magnetization and energy in the critical regimeT=T_c of two-dimensional Ising models. The analysis of these fluctuations leads to numerical determination of the critical exponents for dynamics: for the size dependence of correlation time atT _c , and for frequency dependence of the power spectrumS()~ ^–µ . From the finite-size scaling hypothesis, scaling relations are settled which are confirmed by this numerical experiment.