Nonperturbative renormalization-group approach preserving the momentum dependence of correlation functions

Abstract

We present an approximation scheme of the nonperturbative renormalization group that preserves the momentum dependence of correlation functions. This approximation scheme can be seen as a simple improvement of the local potential approximation (LPA) where the derivative terms in the effective action are promoted to arbitrary momentum-dependent functions. As in the LPA the only field dependence comes from the effective potential, which allows us to solve the renormalization-group equations at a relatively modest numerical cost (as compared, e.g., to the Blaizot-Mendéz-Galain-Wschebor approximation scheme). As an application we consider the two-dimensional quantum O(N ) model at zero temperature. We discuss not only the two-point correlation function but also higher-order correlation functions such as the scalar susceptibility (which allows for an investigation of the "Higgs" amplitude mode) and the conductivity. In particular we show how, using Padé approximants to perform the analytic continuation iωn → ω + i0 + of imaginary frequency correlation functions χ(iωn) computed numerically from the renormalization-group equations, one can obtain spectral functions in the real-frequency domain.

The nonperturbative functional renormalization group and its applications

“…Results for the critical exponents of the three-dimensional O(N) universality class obtained from the LPA and the DE to second, fourth [117,141,145] and sixth [118] orders are shown in Table 1 for N = 0, 1, 2, 3, 4 and compared to Monte Carlo simulations, fixed-dimension perturbative RG, -expansion and conformal bootstrap (the two-Table 1: Critical exponents ν, η and ω for the three-dimensional O(N) universality class obtained in the FRG approach from DE to second [115,116], fourth [117] and sixth [118] orders, LPA [119,120] and BMW approximation [121,122], compared to Monte Carlo (MC) simulations [123][124][125][126][127][128], d = 3 perturbative RG (PT) [14], -expansion at order 6 ( -exp) [129] and conformal bootstrap (CB) [130][131][132][133][134] (when several estimates are available in the literature, we show the one with the smallest error bar). (12) 0.72 0.765 0.774(20) 0.794(9) 0.817 (30) dimensional O(1) model (Ising universality class) is discussed in [146][147][148]).…”

“…(12) 0.72 0.765 0.774(20) 0.794(9) 0.817 (30) dimensional O(1) model (Ising universality class) is discussed in [146][147][148]). In the large-N limit the DE to second order becomes exact for the critical exponents and the functions U k (ρ) and Z k (ρ) [91,94,111,120,149]. 15 Since the DE is a priori valid in all dimensions and for all N, it can be applied to the two-dimensional O(2) model where the transition, as predicted by BKT [30][31][32][33], is driven by topological defects (vortices).…”

“…In some problems, it is necessary to keep both the momentum dependence of low-order vertices and the full set of vertices in the zero-momentum sector (which amounts to considering the full effective potential). A possible approximation [119,120,212], inspired by the LPA , is defined by the effective action…”

Bose-glass phase of a one-dimensional disordered Bose fluid: metastable states, quantum tunneling and droplets

“…In principle one can obtain the full frequency dependence of the conductivity by performing the analytic continuation from the numerically known ∆ k=0 (iω) by means of the resonances-via-Padé method. [84][85][86] This procedure has been successfully used in many works using the FRG formalism, [87][88][89][90][91][92][93] but turns out to be rather unstable in the present case, presumably because of the nonuniform convergence of ∆ k (iω) towards the singular function (68).…”

Self-similar sequence transformation for critical exponents

“…The results of this calculation for the exponent ν are illustrated in Table 1. Following the same procedure for the exponents η and ω, we obtain the values shown in Table 2, where we compare the results obtained by means of factor approximants (FA) with those of other methods: Monte Carlo simulations (MC) [28][29][30][31][32][33], Conformal bootstrap (CB) [34][35][36][37][38], Hypergeometric Meijer summation (HGM) [27], Borel summation complimented by additional conjectures on the behavior of coefficients (BAC) [39], Borel summation with conformal mapping (BCM) [40], and Nonperturbative renormalization group (NPRG) [5,[41][42][43].…”