Scaling laws and surface exponents from renormalization group equations

The critical behaviour of a semi-infinite system with O(n) spin symmetry is studied. Renormalization group equations are derived whose solutions yield scaling laws for surface exponents. Results to order ² are given which disprove the conjecture η_; = ν⁻¹ by Bray and Moore.     

Symmetry breaking and planar bootstrap equations

Symmetry breaking is discussed using the planar bootstrap condition in the form of Rosenzweig and Veneziano. It is suggested that the bootstrap equations with symmetric couplings allow certain non-symmetric (non-degenerate) solutions for the trajectories, in particular, solutions with small SU(3) breaking but with badly broken SU(4) symmetry.     

Scaling of fluctuations and critical exponents

We present a new technique to describe the abnormal behavior of certain fluctuation observables in the critical regime of quantum statistical systems which undergo a phase transition. The idea is to rescale the local fluctuation operators by a relevant external parameter of the system, in addition to the usual scaling with the inverse square root of the volume. The scaling indices used in this scaling procedure are directly related to the critical exponents. Furthermore, it is explained that this new method of scaling preserves the CCR structure of the algebra of macroscopic fluctuations. Finally, scaling indices are computed for the relevant microscopic observables at all temperatures in a mean field approximation for a quantum anharmonic crystal. These indices yield the same critical exponents as predicted by mean field theory.     

High-temperature expansion of surface critical exponents for the classical n-vector model

The surface critical exponents of the classical n-vector model are investigated in the high-temperature series expansion. The first nine coefficients of the layer susceptibility for the semi-infinite simple cubic lattice are presented. The n-dependence of surface exponents is compared with the results of other theories.     

Hypefine critical exponents and probe disturbance effects

The problem of probe disturbance is considered for static critical exponents that are determined from hyperfine fields on isolated impurity probes in a magnetic solid. Based on Ising model calculations and reanalysis of relevant experimental data, it is concluded that for reduced temperatures restricted tot10^–2, the systematic error made in a hyperfine measurement of the exponent is less than 2%.Work supported in part by the National Science Foundation, USA, and the Stichting voor Fundamenteel Onderzoek der Materie, the Netherlands.     

Anomalous critical exponents in the anisotropic Ashkin-Teller model

We perform a rigorous computation of the specific heat of the Ashkin-Teller model in the case of small interaction and we explain how the universality-nonuniversality crossover is realized when the isotropic limit is reached. We prove that, even in the region where universality for the specific heat holds, anomalous critical exponents appear: for instance, we predict the existence of a previously unknown anomalous exponent, continuously varying with the strength of the interaction, describing how the difference between the critical temperatures rescales with the anisotropy parameter.     

Normal form expansion in an algebraic way

It is shown that in an algebraic approach, normal form expansion of operators, bounded or not, but densely defined on Fock space, converge weakly.     

Hyperfine critical exponents β and δ of tantalum in nickel

The hyperfine critical coefficients β and δ on dilute ¹⁸¹Ta nuclei in a nickel matrix have been measured using the time differential perturbed angular correlation technique. We find: β = 0.041₇ ± 0.01₀, δ = 4.1 ± 0.3. They are somewhat different from the values deduced from the bulk magnetization measurements on pure nickel.     

Gauge-invariant critical exponents for the Ginzburg-Landau model

The critical behavior of the Ginzburg-Landau model is described in a manifestly gauge-invariant manner. The gauge-invariant correlation-function exponent is computed to first order in the 4-d and 1/n expansion, and found to agree with the ordinary exponent obtained in the covariant gauge, with the parameter alpha=1-d in the gauge-fixing term ( partial differential (mu)A(mu))(2)/2alpha.     

Spontaneous magnetization from NMR measurements and critical exponents in nickel

Nuclear magnetic resonance of Ni⁶¹ has been studied in pure nickel metal (of natural isotope composition) in the vicinity of the Curie point. The temperature dependence of NMR frequency has been measured up to 0·96T _c. Determination ofT _c was aided by initial susceptibility measurements. The analysis of these results yields =0·336±0·002 for the critical exponent describing the decrease of spontaneous magnetization to zero atT _c. This value of is in excellent agreement with the corresponding values obtained from NMR experiments in ferromagnetic EuS and antiferromagnetic MnF₂, it differs, however, from the values obtained for nickel by other methods. In a general discussion of this problem which uses the theoretical relations between various critical exponents the earlier data, in particular those of Weiss and Forrer, are reexamined.It is a pleasure for the authors to acknowledge the valuable advice and helpful comments of Professor J. Bro and Dr. J. ternberk. They are indebted also to Dr. V. Houdek for help with the susceptibility measurements, which were performed on his apparatus.     

Evaluation of critical exponents from raman intensities

Theory and experiment are compared for Raman intensities near continuous structural phase transitions. Situations in which the order parameter couples linearly to light or quadratically are considered. Both cases are easily analyzed for the soft modes in ferroelastic LnP₅O₁₄ (Ln = La, Pr, Nd, Tb). The trace polarizability tensor yields β = 0.49 ± 0.02; off-diagonal terms give γ’ = 1.16 ± 0.15 and γ = 1.07 ± 0.10. Mean field results are also obtained for barium sodium niobate near T (incommensurate) = 582 K and for tris-sarcosine calcium chloride near T _c = 128 K.     

The critical exponents of crystalline random surfaces

We report on a high statistics numerical study of the crystalline random surface model with extrinsic curvature on lattices of up to 64² points. The critical exponents at the crumpling transition are determined by a number of methods all of which are shown to agree within estimated errors. The correlation length exponent is found to be ν = 0.71(5) from the tangent-tangent correlation function whereas we find ν = 0.73(6) by assuming finite size scaling of the specific heat peak and hyperscaling. These results imply a specific heat exponent α = 0.58(10); this is a good fit to the specific heat on a 64² lattice with a χ² per degree of freedom of 1.7 although the best direct fit to the specific heat data yields a much lower value of a. We have measured the normal-normal correlation function in the crumpled phase and find that, within the accuracy of our simulations, the data can be described by a super-renormalizable field theory.     

Statistical bootstrap model and critical behaviour of hadronic matter

The critical behaviour of two different macroscopic hadron systems is studied, using a linearly exponential mass spectrum:p≈-m ^a exp(bm). It is shown that near the critical point, fora相似文献   

Strict inequalities for some critical exponents in two-dimensional percolation

For 2D percolation we slightly improve a result of Chayes and Chayes to the effect that the critical exponent for the percolation probability isstrictly less than 1. The same argument is applied to prove that ifL():={(x, y):x=r cos, y=r sin for some r0, or} and():=limpp _c [log(p–p _c )]^–1 log P_cr {itO is connected to by an occupied path inL()}, then() is strictly decreasing in on [0, 2]. Similarly, lim_n [–logn]^–1 logP _cr {itO is connected by an occupied path inL()() to the exterior of [–n, n]×[–n, n] is strictly decreasing in on [0, 2].