arXiv subjects

Y. Meurice

Publications and source records attributed to Y. Meurice.

89 records · Page 5Linked to original sources

arXiv2000

Non-Gaussian numerical errors versus mass hierarchy

We probe the numerical errors made in renormalization group calculations by varying slightly the rescaling factor of the fields and rescaling back in order to get the same (if there were no round-off errors) zero momentum 2-point function (magnetic susceptibility). The actual calculations were performed with Dyson's hierarchical model and a simplified version of it. We compare the distributions of numerical values obtained from a large sample of rescaling factors with the (Gaussian by design) distribution of a random number generator and find significant departures from the Gaussian behavior. In addition, the average value differ (robustly) from the exact answer by a quantity which is of the same order as the standard deviation. We provide a simple model in which the errors made at shorter distance have a larger weight than those made at larger distance. This model explains in part the non-Gaussian features and why the central-limit theorem does not apply.

hep-lat↗

arXiv1999

Hyperscaling in the Broken Symmetry Phase of Dyson's Hierarchical Model

We use polynomial truncations of the Fourier transform of the local measure to calculate the connected q-point functions of Dyson's hierarchical model in the broken symmetry phase. We show that accurate values of the connected 1, 2 and 3 point functions can be obtained at large volume and in a limited range of constant external field coupled linearly to the field variable. We introduce a new method to obtain the correct infinite volume and zero external field extrapolations. We extract the leading critical exponents and show that they obey the scaling and hyperscaling relations with an accuracy ranging from 10^-5 to 5 10^-3. We briefly discuss how to improve the method of calculation.

hep-lat↗

arXiv1999

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.

hep-lat↗

arXiv1999

Non-Universal Quantities from Dual RG Transformations

Using a simplified version of the RG transformation of Dyson's hierarchical model, we show that one can calculate all the non-universal quantities entering into the scaling laws by combining an expansion about the high-temperature fixed point with a dual expansion about the critical point. The magnetic susceptibility is expressed in terms of two dual quantities transforming covariantly under the RG transformation and has a smooth limit in the high-temperature limit. Using the analogy with hamiltonian systems, the simplified example discussed here is similar to the anharmonic oscillator, while more realistic examples can be tought of as coupled oscillators allowing resonance phenomena.

hep-lat↗

arXiv1998

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.

hep-lat↗

arXiv1998

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.

hep-lat↗

arXiv1998

Accurate Checks of Universality for Dyson's Hierarchical Model

Using recently developed methods, we perform high-accuracy calculations of the susceptibility near beta_c for the D=3 version of Dyson's hierarchical model. Using linear fits, we estimate the leading gamma and subleading Delta exponents. Independent estimates are obtained by calculating the first two eigenvalues of the linearized renormalization group transformation. We found gamma = 1.29914073 (with an estimated error of 10^{-8}) and, Delta=0.4259469 (with an estimated error of 10^{-7}) independently of the choice of local integration measure (Ising or Landau-Ginzburg). After a suitable rescaling, the approximate fixed points for a large class of local measure coincide accurately with a fixed point constructed by Koch and Wittwer.

hep-lat↗

arXiv1997

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 hyperscaling

hep-lat↗

arXiv1997

A Guide to Precision Calculations in Dyson's Hierarchical Scalar Field Theory

The goal of this article is to provide a practical method to calculate, in a scalar theory, accurate numerical values of the renormalized quantities which could be used to test any kind of approximate calculation. We use finite truncations of the Fourier transform of the recursion formula for Dyson's hierarchical model in the symmetric phase to perform high-precision calculations of the unsubtracted Green's functions at zero momentum in dimension 3, 4, and 5. We use the well-known correspondence between statistical mechanics and field theory in which the large cut-off limit is obtained by letting beta reach a critical value beta_c (with up to 16 significant digits in our actual calculations). We show that the round-off errors on the magnetic susceptibility grow like (beta_c -beta)^{-1} near criticality. We show that the systematic errors (finite truncations and volume) can be controlled with an exponential precision and reduced to a level lower than the numerical errors. We justify the use of the truncation for calculations of the high-temperature expansion. We calculate the dimensionless renormalized coupling constant corresponding to the 4-point function and show that when beta -> beta_c, this quantity tends to a fixed value which can be determined accurately when D=3 (hyperscaling holds), and goes to zero like (Ln(beta_c -beta))^{-1} when D=4.

hep-lat↗

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↗

arXiv1995

Precise Determination of the Energy Levels of the Anharmonic Oscillator from the Quantization of the Angle Variable

Using an ansatz motivated by the classical form of $e^{iϕ}$, where $ϕ$ is the angle variable, we construct operators which satisfy the commutation relations of the creation-annihilation operators for the anharmonic oscillator. The matrix elements of these operators can be expressed in terms of {\it entire} functions in the position complex plane. These functions provide solutions of the Ricatti equation associated with the time-independent Schrödinger equation. We relate the normalizability of the eigenstates to the global properties of the flows of this equation. These exact results yield approximations which complement the WKB approximation and allow an arbitrarily precise determination of the energy levels. We give numerical results for the first 10 levels with 30 digits. We address the question of the quantum integrability of the system.

quant-ph↗

arXiv1994

The High-Temperature Expansion of the Hierarchical Ising Model: From Poincaré Symmetry to an Algebraic Algorithm

We show that the hierarchical model at finite volume has a symmetry group which can be decomposed into rotations and translations as the familiar Poincaré groups. Using these symmetries, we show that the intricate sums appearing in the calculation of the high-temperature expansion of the magnetic susceptibility can be performed, at least up to the fourth order, using elementary algebraic manipulations which can be implemented with a computer. These symmetries appear more clearly if we use the 2-adic fractions to label the sites. We then apply the new algebraic methods to the calculation of quantities having a random walk interpretation. In particular, we show that the probability of returning at the starting point after $m$ steps has poles at $D=-2,-4,....-2m$ , where $D$ is a free parameter playing a role similar to the dimensionality in nearest neighbor models.

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↗