The magnetized (2+1)-dimensional Gross-Neveu model at finite density

We perform a lattice study of the (\(2+1\))-dimensional Gross-Neveu model in a background magnetic field \(B\) and at non-zero chemical potential \(\mu\). The complex-action problem arising in our simulations using overlap fermions is under control. For \(B=0\) we observe a first-order phase transition in \(\mu\) even at non-vanishing temperatures. Our main finding, however, is that the rich phase structure found in the limit of infinite flavor number \(N_\mathrm{f}\) is washed out by the fluctuations present at \(N_\mathrm{f}=1\). We find no evidence for inverse magnetic catalysis, i.e., the decrease of the order parameter of chiral symmetry breaking with \(B\) for \(\mu\) close to the chiral phase transition. Instead, the magnetic field tends to enhance the breakdown of chiral symmetry for all values of \(\mu\) below the transition. Moreover, we find no trace of spatial inhomogeneities in the order parameter. We briefly comment on the potential relevance of our results for QCD.