arXiv subjects

G. Ordaz

Publications and source records attributed to G. Ordaz.

6 recordsLinked to original sources

arXiv1996

Non-Perturbative Aspects of Scalar Field theory

Using the hierarchical approximation, we discuss the cut-off dependence of the renormalized quantities of a scalar field theory. The naturalness problem and questions related to triviality bounds are briefly discussed. We discuss unphysical features associated with the hierarchical approximation such as the recently observed oscillatory corrections to the scaling laws. We mention a two-parameter family of recursion formulas which allows one to continuously extrapolate between Wilson's approximate recursion formula and the recursion formula of Dyson's hierarchical model. The parameters are the dimension D and 2^zeta, the number of sites integrated in one RG transformation. We show numerically that at fixed D, the critical exponent gamma depends continuously on zeta. We suggest the requirement of zeta -independence as a guide for constructing improved recursion formulas.

hep-th↗

arXiv1996

The Oscillatory Behavior of the High-Temperature Expansion of Dyson's Hierarchical Model: A Renormalization Group Analysis

We calculate 800 coefficients of the high-temperature expansion of the magnetic susceptibility of Dyson's hierarchical model with a Landau-Ginzburg measure. Log-periodic corrections to the scaling laws appear as in the case of a Ising measure. The period of oscillation appears to be a universal quantity given in good approximation by the logarithm of the largest eigenvalue of the linearized RG transformation, in agreement with a possibility suggested by K. Wilson and developed by Niemeijer and van Leeuwen. We estimate $γ$ to be 1.300 (with a systematic error of the order of 0.002) in good agreement with the results obtained with other methods such as the $ε$-expansion. We briefly discuss the relationship between the oscillations and the zeros of the partition function near the critical point in the complex temperature plane.

hep-lat↗

arXiv1996

A Two-Parameter Recursion Formula For Scalar Field Theory

We present a two-parameter family of recursion formulas for scalar field theory. The first parameter is the dimension $(D)$. The second parameter ($ζ$) allows one to continuously extrapolate between Wilson's approximate recursion formula and the recursion formula of Dyson's hierarchical model. We show numerically that at fixed $D$, the critical exponent $γ$ depends continuously on $ζ$. We suggest the use of the $ζ-$independence as a guide to construct improved recursion formulas.

hep-lat↗

arXiv1995

Evidence for Complex Subleading Exponents from the High-Temperature Expansion of the Hierarchical Ising Model

Using a renormalization group method, we calculate 800 high-temperature coefficients of the magnetic susceptibility of the hierarchical Ising model. The conventional quantities obtained from differences of ratios of coefficients show unexpected smooth oscillations with a period growing logarithmically and can be fitted assuming corrections to the scaling laws with complex exponents.

hep-lat↗

arXiv1995

The Elusive Asymptotic Behavior of the High-Temperature Expansion of the Hierarchical Ising Model

We present a differential formulation of the recursion formula of the hierarchical model which provides a recursive method of calculation for the high-temperature expansion. We calculate the first 30 coefficients of the high temperature expansion of the magnetic susceptibility of the Ising hierarchical model with 12 significant digits. We study the departure from the approximation which consists of identifying the coefficients with the values they would take if a $[0,1]$ Padé approximant were exact. We show that, when the order in the high-temperature expansion increases, the departure from this approximation grows more slowly than for nearest neighbor models. As a consequence, the value of the critical exponent $γ$ estimated using Padé approximants converges very slowly and the estimations using 30 coefficients have errors larger than 0.05. A (presumably much) larger number of coefficients is necessary to obtain the critical exponents with a precision comparable to the precision obtained for nearest neighbor models with less coefficients. We also discuss the possibility of constructing models where a $[0,1]$ Padé approximant would be exact.

hep-lat↗

arXiv1994

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 ($σ=\pm 1$) 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 $2^18$ sites have been used. Surprisingly, the numerical data can be fitted very well with a simple power law of the form $(1- β/β_c )^{- γ} $for the {\it whole} temperature range. The numerical values for $γ$ agree within a few percent with the values calculated with a high-temperature expansion but show significant discrepancies with the epsilon-expansion. We would appreciate comments about these results.

hep-lat↗