Renormalization Group for Mixed Fermion-Boson Systems

TL;DR

This paper develops a momentum-shell renormalization group method for systems with both fermions and bosons, especially near forward scattering, addressing phase space constraints and different dynamic exponents, applicable to condensed matter problems.

Contribution

It introduces a versatile RG framework for mixed fermion-boson systems with near-forward scattering, handling various dynamic exponents and phase space constraints.

Findings

Equivalent results for z=1 using multiple formalisms

A consistent RG scheme for z≠1 cases

Applicable to condensed matter systems like magnets and gauge fields

Abstract

We formulate a momentum-shell renormalization group (RG) procedure that can be used in theories containing both bosons and fermions with a Fermi surface. We focus on boson-fermion couplings that are nearly forward-scattering, {\it i.e.} involving small momentum transfer ($\vec{q} \approx 0$) for the fermions. Special consideration is given to phase space constraints that result from the conservation of momentum and the imposition of ultraviolet cutoffs. For problems where the energy and momentum scale similarly (dynamic exponent $z = 1$), we show that more than one formalism can be used and they give equivalent results. When the energy and momentum must scale differently ($z \neq 1$), the procedures available are more limited but a consistent RG scheme can still be formulated. Our approach is applicable to a variety of problems in condensed matter physics, such as itinerant-electron magnets and gauge fields interacting with fermions.