Emergence of a control parameter for the antiferromagnetic quantum critical metal

We study the antiferromagnetic quantum critical metal in 3 - is an element of 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 is an element of 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 fixed point, and the is an element of 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 is an element of in the limit that the ratio of velocities vanishes. The is an element of expansion gives critical exponents which are consistent with the exact solution obtained in 0 < is an element of <= 1.