Symmetry Breaking, Duality and Fine-Tuning in Hierarchical Spin Models

We discuss three questions related to the critical behavior of hierarchical spin models: 1) the hyperscaling relations in the broken symmetry phase; 2) the combined use of dual expansions to calculate the non-universal quantities; 3) the fine-tuning issue in approximately supersymmetric models.Comment: 3 pages, 1 figure, Lattice99 (spin

Universality, Scaling and Triviality in a Hierarchical Field Theory

Using polynomial truncations of the Fourier transform of the RG transformation of Dyson's hierarchical model, we show that it is possible to calculate very accurately the renormalized quantities in the symmetric phase. Numerical results regarding the corrections to the scaling laws, (i.e finite cut-off dependence) triviality, hyperscaling, universality and high-accuracy determinations of the critical exponents are discussed.Comment: LATTICE98(spin

High-Accuracy Calculations of the Critical Exponents of Dyson's Hierarchical Model

We calculate the critical exponent gamma of Dyson's hierarchical model by direct fits of the zero momentum two-point function, calculated with an Ising and a Landau-Ginzburg measure, and by linearization about the Koch-Wittwer fixed point. We find gamma= 1.299140730159 plus or minus 10^(-12). We extract three types of subleading corrections (in other words, a parametrization of the way the two-point function depends on the cutoff) from the fits and check the value of the first subleading exponent from the linearized procedure. We suggest that all the non-universal quantities entering the subleading corrections can be calculated systematically from the non-linear contributions about the fixed point and that this procedure would provide an alternative way to introduce the bare parameters in a field theory model.Comment: 15 pages, 9 figures, uses revte

A Check of a D=4 Field-Theoretical Calculation Using the High-Temperature Expansion for Dyson's Hierarchical Model

We calculate the high-temperature expansion of the 2-point function up to order 800 in beta. We show that estimations of the critical exponent gamma based on asymptotic analysis are not very accurate in presence of confluent logarithmic singularities. Using a direct comparison between the actual series and the series obtained from a parametrization of the form (beta_c -beta)^(-gamma) (Ln(beta_c -beta))^p +r), we show that the errors are minimized for gamma =0.9997 and p=0.3351, in very good agreement with field-theoretical calculations. We briefly discuss the related questions of triviality and hyperscalingComment: Uses Revtex, 27 pages including 13 figure