Noncommutativity between the low-energy limit and integer dimension limits in the $\boldsymbol{\epsilon}$-expansion: a case study of the antiferromagnetic quantum critical metal

We study the field theory for the SU($N_c$) symmetric antiferromagnetic quantum critical metal with a one-dimensional Fermi surface embedded in general space dimensions between two and three. The asymptotically exact solution valid in this dimensional range provides an interpolation between the perturbative solution obtained from the $\epsilon$-expansion near three dimensions and the nonperturbative solution in two dimensions. We show that critical exponents are smooth functions of the space dimension. However, physical observables exhibit subtle crossovers that make it hard to access subleading scaling behaviors in two dimensions from the low-energy solution obtained above two dimensions. These crossovers give rise to noncommutativities, where the low-energy limit does not commute with the limits in which the physical dimensions are approached.