# Emergence of a control parameter for the antiferromagnetic quantum critical metal

## TL;DR

This paper extends the analysis of antiferromagnetic quantum critical metals in fractional dimensions, revealing a new control parameter and showing that critical exponents are stable beyond linear order in epsilon.

## Contribution

It introduces a higher-loop analysis showing the importance of two-loop graphs and identifies a velocity ratio as a key small parameter controlling the IR fixed point.

## Key findings

- Two-loop graphs are as important as one-loop due to infrared singularity.
- A velocity ratio emerges as a small parameter suppressing certain diagrams.
- Critical exponents remain unchanged beyond linear order in epsilon when the velocity ratio vanishes.

## Abstract

We study the antiferromagnetic quantum critical metal in $3-\epsilon$ space dimensions by extending the earlier one-loop analysis [Sur and Lee, Phys. Rev. B 91, 125136 (2015)] to higher-loop orders. We show that the $\epsilon$-expansion is not organized by the standard loop expansion, and a two-loop graph becomes as important as one-loop graphs due to an infrared singularity caused by an emergent quasilocality. This qualitatively changes the nature of the infrared (IR) fixed point, and the $\epsilon$-expansion is controlled only after the two-loop effect is taken into account. Furthermore, we show that a ratio between velocities emerges as a small parameter, which suppresses a large class of diagrams. We show that the critical exponents do not receive corrections beyond the linear order in $\epsilon$ in the limit that the ratio of velocities vanishes. The $\epsilon$-expansion gives critical exponents which are consistent with the exact solution obtained in $0 < \epsilon \leq 1$.