Functional renormalization group in the broken symmetry phase: momentum dependence and two-parameter scaling of the self-energy

TL;DR

This paper develops a functional renormalization group approach incorporating spontaneous symmetry breaking to accurately compute the momentum-dependent self-energy in Ginzburg-Landau theories, capturing both perturbative and critical regimes.

Contribution

It introduces a novel RG truncation that includes an order parameter flow equation, enabling detailed analysis of the self-energy's momentum dependence in the broken symmetry phase.

Findings

Provides the full momentum dependence of the self-energy Sigma(k).

Derives a two-parameter scaling function for the self-energy near criticality.

Interpolates between perturbative and critical scaling regimes.

Abstract

We include spontaneous symmetry breaking into the functional renormalization group (RG) equations for the irreducible vertices of Ginzburg-Landau theories by augmenting these equations by a flow equation for the order parameter, which is determined from the requirement that at each RG step the vertex with one external leg vanishes identically. Using this strategy, we propose a simple truncation of the coupled RG flow equations for the vertices in the broken symmetry phase of the Ising universality class in D dimensions. Our truncation yields the full momentum dependence of the self-energy Sigma (k) and interpolates between lowest order perturbation theory at large momenta k and the critical scaling regime for small k. Close to the critical point, our method yields the self-energy in the scaling form Sigma (k) = k_c^2 sigma^{-} (k | xi, k / k_c), where xi is the order parameter correlation length, k_c is the Ginzburg scale, and sigma^{-} (x, y) is a dimensionless two-parameter scaling function for the broken symmetry phase which we explicitly calculate within our truncation.