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

TL;DR

This study investigates whether the Gross-Neveu and Thirring models are equivalent at N=1 across different regimes, confirming equivalence at zero density but finding discrepancies at finite chemical potential using two-loop optimized perturbation theory.

Contribution

The paper provides a detailed two-loop analysis of the planar Gross-Neveu and Thirring models at N=1, revealing conditions under which their equivalence holds or breaks down.

Findings

Models are thermodynamically equivalent at zero density for N=1.

Equivalence breaks down at finite chemical potential.

Identified contributions causing the discrepancy.

Abstract

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, $N=1$. 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 $N=1$. However, at finite chemical potentials we find that the $N=1$ Fierz equivalence no longer holds. After examining the relevant free energies, 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.