Testing the equivalence between the planar Gross-Neveu and Thirring models at N=1

It is known that the Fierz identities predict that the Gross-Neveu and Thirring models should be equivalent when describing systems composed of a single fermionic flavor, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>N</a:mi><a:mo>=</a:mo><a:mn>1</a:mn></a:math>. Here, we consider the planar version of both models within the framework of the optimized perturbation theory at the two-loop level, in order to verify if the predicted equivalence emerges explicitly when different temperature and density regimes are considered. At vanishing densities, our results indicate that both models indeed describe exactly the same thermodynamics, provided that <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>N</c:mi><c:mo>=</c:mo><c:mn>1</c:mn></c:math>. However, at finite chemical potentials we find that the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>N</e:mi><e:mo>=</e:mo><e:mn>1</e:mn></e:math> Fierz equivalence no longer holds. After examining the relevant thermodynamic potentials, we have identified the contributions which lead to this puzzling discrepancy. Finally, we discuss different frameworks in which this (so far open) problem could be further understood and eventually circumvented. Published by the American Physical Society 2024