Chiral Symmetry Restoration and Realisation of the Goldstone Mechanism in the U(1) Gross-Neveu Model at Non-Zero Chemical Potential

We simulate the Gross-Neveu model in 2+1 dimensions at nonzero baryon density (chemical potential mu =/= 0). It is possible to formulate this model with a real action and therefore to perform standard hybrid Monte Carlo simulations with mu =/= 0 in the functional measure. We compare the physical observables from these simulations with simulations using the Glasgow method where the value of mu in the functional measure is fixed at a value mu_upd. We find that the observables are sensitive to the choice of mu_upd. We consider the implications of our findings for Glasgow method QCD simulations at mu =/= 0. We demonstrate that the realisation of the Goldstone mechanism in the Gross-Neveu model is fundamentally different from that in QCD. We find that this difference explains why there is an unphysical transition in QCD simulations at mu =/= 0 associated with the pion mass scale whereas the transition in the Gross-Neveu model occurs at a larger mass scale and is therefore consistent with theoretical predictions. We note classes of theories which are exceptions to the Vafa-Witten theorem which permit the possibility of formation of baryon number violating diquark condensates.