Functional renormalization group approach to the Ising-nematic quantum critical point of two-dimensional metals

TL;DR

This paper applies the functional renormalization group to study the Ising-nematic quantum critical point in 2D metals, explicitly calculating interaction effects and vertex corrections beyond one-loop approximations.

Contribution

It introduces a simplified truncation of the FRG flow equations that captures three-loop diagram effects and computes the fermion anomalous dimension.

Findings

Three-loop diagrams are included in the FRG truncation.

Vertex corrections significantly affect the fermion anomalous dimension.

The approach provides a more complete understanding of quantum critical behavior.

Abstract

Using functional renormalization group methods, we study an effective low-energy model describing the Ising-nematic quantum critical point in two-dimensional metals. We treat both gapless fermionic and bosonic degrees of freedom on equal footing and explicitly calculate the momentum and frequency dependent effective interaction between the fermions mediated by the bosonic fluctuations. Following earlier work by S.-S. Lee for a one-patch model, Metlitski and Sachdev [Phys. Rev. B {\bf{82}}, 075127] recently found within a field-theoretical approach that certain three-loop diagrams strongly modify the one-loop results, and that the conventional 1/N expansion breaks down in this problem. We show that the singular three-loop diagrams considered by Metlitski and Sachdev are included in a rather simple truncation of the functional renormalization group flow equations for this model involving only irreducible vertices with two and three external legs. Our approximate solution of these flow equations explicitly yields the vertex corrections of this problem and allows us to calculate the anomalous dimension $\eta_{\psi}$ of the fermion field.