Fermion loops and improved power-counting in two-dimensional critical metals with singular forward scattering

TL;DR

This paper investigates the perturbation expansion in two-dimensional quantum critical metals with singular forward scattering, revealing cancellations that improve divergence estimates and show convergence properties depending on the dynamical critical exponent.

Contribution

It derives asymptotic properties of fermion loops and demonstrates how cancellations improve power-counting, clarifying divergence behavior in these critical metals.

Findings

Perturbative contributions to boson self-energy are UV convergent for z<3.

Cancellations reduce divergence degrees in Feynman diagrams.

Divergences occur beyond three-loop order for z≥3.

Abstract

We analyze general properties of the perturbation expansion for two-dimensional quantum critical metals with singular forward scattering, such as metals at an Ising nematic quantum critical point and metals coupled to a U(1) gauge field. We derive asymptotic properties of fermion loops appearing as subdiagrams of the contributing Feynman diagrams -- for large and small momenta. Substantial cancellations are found in important scaling limits, which reduce the degree of divergence of Feynman diagrams with boson legs. Implementing these cancellations we obtain improved power-counting estimates that yield the true degree of divergence. In particular, we find that perturbative contributions to the boson self-energy are generally ultraviolet convergent for a dynamical critical exponent $z<3$, and divergent beyond three-loop order for $z \geq 3$.