Functional renormalization group with vacuum expectation values and spontaneous symmetry breaking

TL;DR

This paper extends the super-field functional renormalization group method to include vacuum expectation values, enabling the study of spontaneous symmetry breaking and collective phenomena in interacting Fermi systems.

Contribution

The authors develop a generalized RG framework that incorporates finite vacuum expectation values, allowing for the analysis of symmetry breaking within a unified functional approach.

Findings

Derived exact hierarchy of flow equations for vertices and vacuum expectation values.

Applied the method to an interacting Fermi system with a collective bosonic field.

Connected the bosonic vacuum expectation value to fermionic density and calculated compressibility.

Abstract

We generalize our recently developed super-field functional renormalization group (RG) method involving both Fermi and Bose fields [F. Schuetz, L. Bartosch, and P. Kopietz, Phys. Rev. B 72, 035105 (2005)] to include the possibility that some bosonic components of the field have a finite vacuum expectation value. We derive an exact hierarchy of flow equations for the one-line irreducible vertices and the vacuum expectation value of the field. We apply our method to an interacting Fermi system where the interaction can be decoupled in the zero-sound channel and is then mediated by a collective bosonic field. The vacuum expectation value of the zero-frequency and zero-momentum component of the bosonic field is then closely related to the fermionic density. This can be exploited to calculate the compressibility of the interacting system. By using a cutoff in the bosonic propagator, the RG can be set up such that the self-consistent Hartree approximation is imposed as initial condition for the RG flow and the corrections to this approximation are generated as the remaining degrees of freedom are successively eliminated by the RG procedure.