arXiv subjects

Y. Meurice

Publications and source records attributed to Y. Meurice.

At least 73 records · Page 4Linked to original sources

arXiv2005

Asymptotically Universal Crossover in Perturbation Theory with a Field Cutoff

We discuss the crossover between the small and large field cutoff (denoted x_{max}) limits of the perturbative coefficients for a simple integral and the anharmonic oscillator. We show that in the limit where the order k of the perturbative coefficient a_k(x_{max}) becomes large and for x_{max} in the crossover region, a_k(x_{max}) is proportional to the integral from -infinity to x_{max} of e^{-A(x-x_0(k))^2}dx. The constant A and the function x_0(k) are determined empirically and compared with exact (for the integral) and approximate (for the anharmonic oscillator) calculations. We discuss how this approach could be relevant for the question of interpolation between renormalization group fixed points.

hep-th↗

arXiv2005

An example of optimal field cut in lattice gauge perturbation theory

We discuss the weak coupling expansion of a one plaquette SU(2) lattice gauge theory. We show that the conventional perturbative series for the partition function has a zero radius of convergence and is asymptotic. The average plaquette is discontinuous at g^2=0. However, the fact that SU(2) is compact provides a perturbative sum that converges toward the correct answer for positive g^2. This alternate methods amounts to introducing a specific coupling dependent field cut, that turns the coefficients into g-dependent quantities. Generalizing to an arbitrary field cut, we obtain a regular power series with a finite radius of convergence. At any order in the modified perturbative procedure, and for a given coupling, it is possible to find at least one (and sometimes two) values of the field cut that provide the exact answer. This optimal field cut can be determined approximately using the strong coupling expansion. This allows us to interpolate accurately between the weak and strong coupling regions. We discuss the extension of the method to lattice gauge theory on a D-dimensional cubic lattice.

hep-lat↗

arXiv2005

Universal Crossover in Perturbation Theory with a Large Field Cutoff

For lambda phi^4 models, the introduction of a large field cutoff improves significantly the accuracy that can be reached with perturbative series but the calculation of the modified coefficients remains a challenging problem. We show that this problem can be solved numerically, and in the limits of large and small field cutoffs, for the ground state energy of the anharmonic oscillator. For the two lowest orders, the approximate formulas obtained in the large field cutoff limit extend unexpectedly far in the low field cutoff region. For the higher orders, the transition between the small field cutoff regime and the large field cutoff regime can be described in terms of a universal function.

hep-th↗

arXiv2004

Large Field Cutoffs in Lattice Gauge Theory

In pure gauge SU(3) near beta = 6, weak and strong coupling expansions break down and the MC method seems to be the only practical alternative. We discuss the possibility of using a modified version of perturbation theory which relies on a large field cutoff and has been successfully applied to the double-well potential (Y. M., PRL 88 141601). Generically, in the case of scalar field theory, the weak coupling expansion is unable to reproduce the exponential suppression of the large field configurations. This problem can be solved by introducing a large field cutoff. The value of this cutoff can be chosen to reduce the discrepancy with the original problem. This optimization can be approximately performed using the strong coupling expansion and bridges the gap between the two expansions. We report recent attempts to extend this procedure for SU(3) gauge theory on the lattice. We compare gauge invariant and gauge dependent (in the Landau gauge) criteria to sort the configurations into ``large-field'' and ``small-field'' configurations. %We discuss the effects of discarding the large field configurations. We discuss the convergence of lattice perturbation theory and the way it can be modified in order to obtain results similar to the scalar case.

hep-lat↗

arXiv2004

Lattice gluodynamics at negative g^2

We consider Wilson's SU(N) lattice gauge theory (without fermions) at negative values of beta= 2N/g^2 and for N=2 or 3. We show that in the limit beta -> -infinity, the path integral is dominated by configurations where links variables are set to a nontrivial element of the center on selected non intersecting lines. For N=2, these configurations can be characterized by a unique gauge invariant set of variables, while for N=3 a multiplicity growing with the volume as the number of configurations of an Ising model is observed. In general, there is a discontinuity in the average plaquette when g^2 changes its sign which prevents us from having a convergent series in g^2 for this quantity. For N=2, a change of variables relates the gauge invariant observables at positive and negative values of beta. For N=3, we derive an identity relating the observables at beta with those at beta rotated by +- 2pi/3 in the complex plane and show numerical evidence for a Ising like first order phase transition near beta=-22. We discuss the possibility of having lines of first order phase transitions ending at a second order phase transition in an extended bare parameter space.

hep-lat↗

arXiv2004

Effects of large field cutoffs in scalar and gauge models

We discuss the notion of a large field cutoff for lattice gauge models with compact groups. We propose and compare gauge invariant and gauge dependent (in the Landau gauge) criteria to sort the configurations into ``large-field'' and ``small-field'' configurations. We show that the correlations between volume average of field size indicators and the behavior of the tail of the distribution are very different in the gauge and scalar cases. We show that the effect of discarding the large field configurations on the plaquette average is very different above, below and near beta=5.6 for a pure SU(3) LGT.

hep-lat↗

arXiv2004

Universality in nontrivial continuum limits: a model calculation

We study numerically the continuum limit corresponding to the non-trivial fixed point of Dyson's hierarchical model. We discuss the possibility of using the critical amplitudes as input parameters. We determine numerically the leading and subleading critical amplitudes of the zero-momentum $2l$-point functions in the symmetric phase up to l=10 for randomly chosen local measures. In the infinite cutoff limit, the dimensionless renormalized coupling constants are in very good approximation universal (independent of the choice of the local measure). In addition, ratios of subleading amplitudes also appear to be universal. If we neglect very small log-periodic corrections, the non-universal features of the 2l-point functions appear to depend only the non-universal features of the 2-point function. We infer that when 2l becomes large, the dimensionless renormalized couplings grow as (2l)! despite the non-perturbative nature of our calculation, while the universal ratios of subleading amplitudes grow linearly.

hep-th↗

arXiv2003

Small numerators canceling small denominators of the high-temperature scaling variables: a systematic explanation in arbitrary dimensions

We describe a method to express the susceptibility and higher derivatives of the free energy in terms of the scaling variables (Wegner's nonlinear scaling fields) associated with the high-temperature (HT) fixed point of Dyson hierarchical model in arbitrary dimensions. We give a closed form solution of the linearized problem. We check that up to order 7 in the HT expansion, all the poles ("small denominators") that would naively appear in some positive dimension are canceled by zeroes ("small numerators"). The requirement of continuity in the dimension can be used to lift ambiguities which appear in calculations at fixed dimension. We show that the existence of a HT phase in the infinite volume limit for a continuous set of values of the dimension, requires that this mechanism works to all orders. On the other hand, most poles at negative values of the dimensional parameter (where the free energy density is not well-defined, but RG flows can be studied) persist and reflect the fact that for special negative values of the dimension, finite-size corrections become leading terms. We show that the inverse problem is also free of small denominator problems and that the initial values of the scaling variables can be expressed in terms of the infinite volume limit of the susceptibility and higher derivatives of the free energy. We discuss the existence of an infinite number of conserved quantities (RG invariants) and their relevance for the calculation of universal ratios of critical amplitudes.

cond-mat.stat-mech↗

arXiv2003

The continuum limit of perturbative coefficients calculated with a large field cutoff

We report MC calculations of perturbative coefficients for lattice scalar field theory in dimensions 1, 2 and 3, where the large field contributions are cutoff. This produces converging (instead of asymptotic) perturbative series. We discuss the statistical errors and the lattice effects and show that accurate calculations are possible even in a crossover region where no approximation works. We show that the field cutoff is also a UV regulator. We point out the relevance for QCD questions discussed by Tomboulis and Trottier at this conference.

hep-lat↗

arXiv2003

New Optimization Methods for Converging Perturbative Series with a Field Cutoff

We take advantage of the fact that in lambda phi ^4 problems a large field cutoff phi_max makes perturbative series converge toward values exponentially close to the exact values, to make optimal choices of phi_max. For perturbative series terminated at even order, it is in principle possible to adjust phi_max in order to obtain the exact result. For perturbative series terminated at odd order, the error can only be minimized. It is however possible to introduce a mass shift in order to obtain the exact result. We discuss weak and strong coupling methods to determine the unknown parameters. The numerical calculations in this article have been performed with a simple integral with one variable. We give arguments indicating that the qualitative features observed should extend to quantum mechanics and quantum field theory. We found that optimization at even order is more efficient that at odd order. We compare our methods with the linear delta-expansion (LDE) (combined with the principle of minimal sensitivity) which provides an upper envelope of for the accuracy curves of various Pade and Pade-Borel approximants. Our optimization method performs better than the LDE at strong and intermediate coupling, but not at weak coupling where it appears less robust and subject to further improvements. We also show that it is possible to fix the arbitrary parameter appearing in the LDE using the strong coupling expansion, in order to get accuracies comparable to ours.

hep-th↗

arXiv2002

A study of large field configurations in MC simulations

We discuss a new approach of scalar field theory where the small field contributions are treated perturbatively and the large field configurations (which are responsible for the asymptotic behavior of the perturbative series) are neglected. In two Borel summable lambda phi ^4 problems improved perturbative series can be obtained by this procedure. The modified series converge towards values exponentially close to the exact ones. For lambda larger than some critical value, the method outperforms Pade approximants and Borel summations. The method can also be used for series which are not Borel summable such as the double-well potential series and provide a perturbative approach of the instanton contribution. Semi-classical methods can be used to calculate the modified Feynman rules, estimate the error and optimize the field cutoff. We discuss Monte Carlo simulations in one and two dimensions which support the hypothesis of dilution of large field configurations used in these semi-classical calculations. We show that Monte Carlo methods can be used to calculate the modified perturbative series.

hep-lat↗

arXiv2002

Arbitrarily Accurate Eigenvalues for General Anharmonic Potentials

We show that the Riccati form of the Schrodinger equation can be reformulated in terms of two linear equations depending on an arbitrary function G. When $G$ and the potential are polynomials, the solutions of these two equations are entire functions (L and K) and the zeroes of K are identical to those of the wave function. Requiring such a zero at a large but finite value of the argument yields the low energy eigenstates with exponentially small errors. Judicious choice of G can improve dramatically the numerical treatment. The method yields many significant digits with modest computer means.

quant-ph↗

arXiv2002

Complex singularities of the critical potential in the large-N limit

We show with two numerical examples that the conventional expansion in powers of the field for the critical potential of 3-dimensional O(N) models in the large-N limit, does not converge for values of phi^2 larger than some critical value. This can be explained by the existence of conjugated branch points in the complex phi^2 plane. Pade approximants [L+3/L] for the critical potential apparently converge at large phi^2. This allows high-precision calculation of the fixed point in a more suitable set of coordinates. We argue that the singularities are generic and not an artifact of the large-N limit. We show that ignoring these singularities may lead to inaccurate approximations.

hep-th↗

arXiv2002

From nonlinear scaling fields to critical amplitudes

We propose to combine the nonlinear scaling fields associated with the high-temperature (HT) fixed point, with those associated with the unstable fixed point, in order to calculate the susceptibility and other thermodynamic quantities. The general strategy relies on simple linear relations between the HT scaling fields and the thermodynamic quantities, and the estimation of RG invariants formed out of the two sets of scaling fields. This estimation requires convergent expansions in overlapping domains. If, in addition, the initial values of the scaling fields associated with the unstable fixed point can be calculated from the temperature and the parameters appearing in the microscopic Hamiltonian, one can estimate the critical amplitudes. This strategy has been developed using Dyson's hierarchical model where all the steps can be approximately implemented with good accuracy. We show numerically that for this model (and a simplified version of it), the required overlap apparently occurs, allowing an accurate determination of the critical amplitudes.

cond-mat.stat-mech↗

arXiv2002

A Simple Method to Make Asymptotic Series of Feynman Diagrams Converge

We show that for two non-trivial lambda phi ^4 problems (the anharmonic oscillator and the Landau-Ginzburg hierarchical model), improved perturbative series can be obtained by cutting off the large field contributions. The modified series converge to values exponentially close to the exact ones. For lambda larger than some critical value, the method outperforms Pade's approximants and Borel summations. The method can also be used for series which are not Borel summable such as the double-well potential series. We show that semi-classical methods can be used to calculate the modified Feynman rules, estimate the error and optimize the field cutoff.

hep-th↗

arXiv2000

Testing perturbative results with non-perturbative methods for the Hierarchical model

We present non-perturbative methods to calculate accurately the renormalized quantities for Dyson's Hierarchical Model. We apply this method and calculate the critical exponent gamma with 12 and 4 significant digits in the high and low temperature phases, respectively. We report accurate values for universal ratios of amplitudes and preliminary results concerning the comparison with perturbative results.

hep-lat↗

arXiv2000

Small Numerators Canceling Small Denominators: Is Dyson's Hierarchical Model Solvable?

We present an analytical method to solve Dyson's hierarchical model, involving the scaling variables near the high-temperature fixed point. The procedure seems plagued by the presence of small denominators as in perturbative expansions near integrable systems in Hamiltonian mechanics. However, in all cases considered, a zero denominator always comes with a zero numerator. We conjecture that these cancellations occur in general, suggesting that the model has remarkable features reminiscent of the integrable systems.

cond-mat.stat-mech↗

arXiv2000

Examples of overlapping convergent expansions of scaling variables

We construct series expansions for the scaling variables (which transform multiplicatively under a renormalization group (RG) transformation) in examples where the RG flows, going from an unstable (Wilson's) fixed point to a stable (high-temperature) fixed point, can be calculated numerically. The examples are Dyson's hierarchical model and a simplified version of it. We provide numerical evidence that the scaling variables about the two fixed points have overlapping domain of convergence. We show how quantities such as the magnetic susceptibility can be expressed in terms of these variables. This procedure provide accurate analytical expressions both in the critical and high-temperature region.

hep-lat↗