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Emergence of a control parameter for the antiferromagnetic quantum critical metal

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 quasi-locality. This qualitatively changes the nature of the infrared 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 < 1.