Renormalization Group Operator Research Articles

Manifestly finite perturbation theory for the short-distance expansion of correlation functions in the two-dimensional Ising model

In the spirit of classic works of Wilson on the renormalization group and operator product expansion, a new framework for the study of the theory space of euclidean quantum field theories has been introduced. This formalism is particularly useful for elucidating the structure of the short-distance expansions of the $n$-point functions of a renormalizable quantum field theory near a non-trivial fixed point. We review and apply this formalism in the study of the scaling limit of the two dimensional massive Ising model. Renormalization group analysis and operator product expansions determine all the non-analytic mass dependence of the short-distance expansion of the correlation functions. An extension of the first order variational formula to higher orders provides a manifestly finite scheme for the perturbative calculation of the operator product coefficients to any order in parameters. A perturbative expansion of the correlation functions follows. We implement this scheme for a systematic study of correlation functions involving two spin operators. We show how the necessary non-trivial integrals can be calculated. As two concrete examples we explicitly calculate the short-distance expansion of the spin-spin correlation function to third order and the spin-spin-energy density correlation function to first order in the mass. We also discuss the applicability of our results to perturbations near other non-trivial fixed points corresponding to other unitary minimal models.

Multiscale representation of generating and correlation functions for some models of statistical mechanics and quantum field theory

We consider models of statistical mechanics and quantum field theory (in the Euclidean formulation) which are treated using renormalization group methods and where the action is a small perturbation of a quadratic action. We obtain multiscale formulas for the generating and correlation functions aftern renormalization group transformations which bring out the relation with thenth effective action. We derive and compare the formulas for different RGs. The formulas for correlation functions involve (1) two propagators which are determined by a sequence of approximate wave function renormalization constants and renormalization group operators associated with the decomposition into scales of the quadratic form and (2) field derivatives of the nth effective action. For the case of the block field “δ-function” RG the formulas are especially simple and for asymptotic free theories only the derivatives at zero field are needed; the formulas have been previously used directly to obtain bounds on correlation functions using information obtained from the analysis of effective actions. The simplicity can be traced to an “orthogonality-of-scales” property which follows from an implicit wavelet structure. Other commonly used RGs do not have the “orthogonality of scales” property.

Renormalization group equations in local approximation

For a class of models with Ginzburg-Landau-Wilson functions of a local form it is proved that the spectrum of a renormalization group operator which is linearized near a fixed point is discrete, real, and limited from above. In the framework of a local model, critical exponents for the limitn=∞ are calculated.

Properties of block renormalization group operators for Euclidean Fermions in an external gauge field

The basis is laid for a block renormalization analysis of spinor electrodynamics in d=2,3 dimensions in a lattice approximation. The Fermion part of the renormalization group transformation gives rise to a sequence of localized Fermion action, fluctuation covariance, and minimizer operators in a regular external gauge field. It is shown that these operators are uniformly bounded analytic functions of complex-valued gauge fields with uniform exponentially decaying kernels.

Universal scaling in circle maps

We discuss scaling, renormalization, and universality for critical and subcritical circle maps. Using the structure of the rationals provided by the Farey tree, we construct a pair of renormalization group operators for circle maps with all values of the winding number. In this general formulation, the renormalization equations define a universal invariant set which determines all universal quantities for all cubic maps with all irrational winding numbers. As expected, the manifold of pure rotations defines the subcritical attracting set. We show that the invariant set governing the critical circle maps cannot be embedded in less than 3 dimensions. However, we are able to obtain a differentiable parametrization of the unstable manifold of the critical invariant set, which we believe is universal up to Lipschitz homeomorphisms; the derivative of this function describes the scaling structure of all small gaps in the devil's staircase for all critical maps. Our work can be viewed as providing numerical evidence for Lanford's conjectures on the strange set underlying renormalizations of critical circle maps.

Trajectory scaling function for bifurcations in area-preserving maps on the plane

The trajectory scaling function for area-preserving maps on the plane is found using a calculation of the unstable manifold for the renormalization group operator R· T= Λ· T 2· Λ -1 with Λ= α 0 0 β . Internal self-similarities of high order cycles and of power spectra are deduced.

Pade approximation method as renormalization group method

AbstractIt is shown that the Pade approximation method applied to high temperature series can be regarded as a special renormalization group method. The renormalization group operator is constructured exactly and explicitly.

Renormalization group structure for translationally invariant ferromagnets

We introduce a correlation function description of the renormalization group approach to critical phenomena. Our work is based on treating the renormalization group operator as a linear mapping on the set of Ursell functions, rather than as a nonlinear mapping on the space of Hamiltonians. We mainly consider the ’’mean-spin’’ renormalization group, but the closely related ’’decimation’’ transformation is also considered. Using this approach, we demonstrate for a suitable class of system that the spectrum of the renormalization group operator is bounded and countably infinitely degenerate. We give counterexamples to the notion that there must be convergence to a renormalization group fixed point. Our formulation of the renormalization group is sufficiently general so that convergence to a fixed point does not necessarily imply hyperscaling, i.e., the vanishing of the anomalous dimension of the vacuum, ω*. In the case of convergence to a fixed point we find δ= (d+σ−ω*)/(d−σ−ω*), with ω*?0 necessarily. Above the critical temperature we are able to prove that convergence is obtained to the ’’infinite temperature fixed point,’’ which result generalizes the central limit theorem to ferromagnetically correlated variables.