Emergence of a control parameter for the antiferromagnetic quantum critical metal

We study the antiferromagnetic quantum critical metal in $3-ϵ$ 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 $ϵ$-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 $ϵ$-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 $ϵ$ in the limit that the ratio of velocities vanishes. The $ϵ$-expansion gives critical exponents which are consistent with the exact solution obtained in $0 < \epsilon \leq 1$.