Renormalized Perturbation Theory and Field-Theoretic Renormalization Group

K. B. Efetov

Jayanta K. Bhattacharjee, Deb Shankar Ray

Time-dependent perturbation theory in quantum mechanics is divergent at long times when the perturbation induces a resonance between two eigenstates of the unperturbed Hamiltonian. Divergences in perturbation theory are also common in quantum field theory and in critical phenomena. The renormalization group (RG) was designed to deal with these divergences. In the last two decades, this procedure has been extended to dynamical systems where the perturbation theory diverges in the long-time limit. In this article, we first review the connection between RG in the context of field theory and RG in the context of dynamical systems. We then show that the long-time divergence in the resonant situation in the time-dependent perturbation theory in quantum mechanics can be removed by using a RG-aided calculational scheme.

Michal Hnatič, Matej Kecer, T. Lučivjanský +3 more

In this study we focus on the non-equilibrium phase transition whose asymptotic critical behavior is governed by the dynamic isotropic percolation universality class. In order to quantitatively characterize universal properties associated with this class, we employ a field-theoretic formulation of the model augmented with perturbative renormalization group technique. Utilizing such approach enables us to systematically compute the critical exponents in higher orders of perturbation theory. In particular, we perform an analysis to the three-loop precision. To this order we determine the renormalization constants and present relations for critical exponents in terms of critical dimensions. Finally, we also present numerically estimated values of critical exponents obtained from perturbation theory.

John Cardy

Mehran Kardar A

Achim Rosch, Theo A. Costi, Jens Paaske +1 more

Using a recently developed perturbative renormalization group (RG) scheme, we calculate analytically the spectral function of a Kondo impurity for either large frequencies $\ensuremath{\omega}$ or large magnetic field B and arbitrary frequencies. For large $\ensuremath{\omega}\ensuremath{\gg}\mathrm{max}[{B,T}_{K}]$ the spectral function decays as $1/{\mathrm{ln}}^{2}[\ensuremath{\omega}{/T}_{K}]$ with prefactors which depend on the magnetization. The spin-resolved spectral function displays a pronounced peak at $\ensuremath{\omega}\ensuremath{\sim}B$ with a characteristic asymmetry. In a detailed comparison with results from numerical renormalization group (NRG) and bare perturbation theory in next-to-leading logarithmic order, we show that our perturbative RG scheme is controlled by the small parameter $1/\mathrm{ln}[\mathrm{max}(\ensuremath{\omega}{,B)/T}_{K}].$ Furthermore, we assess the ability of the NRG to resolve structures at finite frequencies.