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

TL;DR

This paper investigates how the order of limits—low-energy and integer dimension—affects the analysis of antiferromagnetic quantum critical metals, revealing noncommutative behaviors that complicate understanding in two dimensions.

Contribution

It provides an interpolating solution between perturbative and nonperturbative regimes across dimensions, highlighting subtle crossovers and noncommutativities in critical behaviors.

Findings

Critical exponents vary smoothly with dimension.

Physical observables show complex crossover behaviors.

Noncommutativity of limits affects low-energy analysis in 2D.

Abstract

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.