Breakdown of the perturbative renormalization group at certain quantum critical points

TL;DR

This paper demonstrates that at certain quantum critical points with multiple time scales, perturbative renormalization group methods break down due to diverging coefficients, challenging their reliability for critical behavior analysis.

Contribution

It reveals the limitations of finite-order perturbative RG approaches at quantum critical points with multiple time scales, highlighting the emergence of non-renormalizable theories.

Findings

Perturbative RG coefficients diverge at specific quantum critical points.

Finite-order perturbative treatments may not approximate true critical behavior.

Quantum ferromagnetic transition in disordered metals exemplifies this breakdown.

Abstract

It is shown that the presence of multiple time scales at a quantum critical point can lead to a breakdown of the loop expansion for critical exponents, since coefficients in the expansion diverge. Consequently, results obtained from finite-order perturbative renormalization-group treatments may be not be an approximation in any sense to the true asymptotic critical behavior. This problem manifests itself as a non-renormalizable field theory, or, equivalently, as the presence of a dangerous irrelevant variable. The quantum ferromagnetic transition in disordered metals provides an example.