Solutions of renormalization-group flow equations with full momentum dependence

Abstract

We demonstrate the power of a recently-proposed approximation scheme for the non-perturbative renormalization group that gives access to correlation functions over their full momentum range. We solve numerically the leading-order flow equations obtained within this scheme, and compute the two-point functions of the O(N ) theories at criticality, in two and three dimensions. Excellent results are obtained for both universal and non-universal quantities at modest numerical cost.PACS numbers: 05.10. Cc,64.60.ae,11.10.Hi The renormalization group, in its non-perturbative version [1,2] (also referred to as the exact renormalization group), provides a general formalism giving access, for arbitrary coupling strength, to a whole set of physically important quantities, universal as well as non-universal [3,4], thermodynamic functions and momentum-dependent correlation functions, etc. However, most studies within this framework involve approximations that restrict their scope to the calculation of thermodynamical quantities, or correlation functions with vanishing external momenta. In order to access the full momentum dependence, Blaizot, Méndez-Galain, and Wschebor (BMW) have recently introduced an approximation scheme which overcomes this limitation [5]. In principle, this scheme allows to compute, in all dimensions, at and away from criticality, both universal and non-universal quantities, as well as momentumdependent properties from p = 0 up to the ultra-violet cut-off Λ (inverse lattice spacing).In this Letter, we present the first complete implementation of the leading order approximation of the BMW scheme, and demonstrate its power by using O(N ) models as a testbed. We compute the entire momentum dependence of the two-point functions in two and three dimensions and obtain excellent results for both universal and non-universal quantities.We start by a brief outline of the formalism. In order to simplify the presentation, we shall write only the equations corresponding to the case of a scalar field theory with quartic coupling, i.e., restrict the presentation to the case N = 1 (corresponding to the Ising model). The strategy of the renormalization group is to build a family of theories indexed by a momentum scale parameter k, such that fluctuations are smoothly taken into account as k is lowered from the microscopic scale Λ down to 0. In practice, this is achieved by adding to the original Euclidean action S a mass-like term of the formis chosen so that R k (q 2 ) ∼ k 2 for q k, which effectively suppresses the modes ϕ(q k), and so that it vanishes for q k, leaving the modes ϕ(q k) unaffected. One then defines a scale-dependent partition functionand a scale-dependent effective actionwith φ = δ ln Z k /δJ. where

Nonperturbative renormalization group preserving full-momentum dependence: Implementation and quantitative evaluation

“…ties [12]. In this section, where we return to the O(N ) models (with general N ), we provide details on the calculation of the critical exponents and check their robustness with respect to variations of the different parameters of the method such as the numerical resolution, the choice of the cut-off function and the location of the normalization point (ρ 0 ,p 0 ).…”

“…In dimension three, the bare coupling constant u has the dimension of a momentum and thus sets a scale (the Ginzburg length: ξ G ∼ u 1 d−4 ). There are typically three momentum domains for Γ (2) (p, ρ = 0) [11,12]:…”

“…In this paper, we provide a full account of the practical implementation of the method without further simplifications. We also detail and extend the first results obtained recently at order s = 2 [12]. In particular, we extract the universal scaling function governing the critical region.…”

“…We thus introduce a renormalization factor Z k , which reflects the finite change of normalization of the field between the ultraviolet scale Λ and the scale k. Within the DE at O(∇ 2 ) this factor describes the overall variation with k of the function Z k (φ) in Eq. (12). We define here Z k by…”

Phase boundary and finite temperature crossovers of the quantum Ising model in two dimensions

“…19 More elaborate truncations lead to improved accuracy in the critical exponents. 14,15,16 From the finite-temperature flows in Fig. 1 (a), we can deduce the Ginzburg-scale, whereũ starts to become sizable, to vary with temperature as Ginzburg scale with the quantum-to-classical crossover scale which follows from the definition ofT in Eq.…”