Mass generation without phase coherence in Gross-Neveu Model with U(1) symmetry at finite temperature and low N. Possible implications for particle physics

The Gross-Neveu model with U(1) symmetry for a small number $N$ of fermions has two characteristic temperatures: $T_{KT}$ that corresponds to onset of quasi-long-range order and $T^* (>T_{KT})$ that corresponds to thermal breaking of fermion pairs. This phenomenon is {\it not} new but is known from the recent studies of superconductors. The region $T_{KT}<T<T^*$ is characterised by a complex gap function $|\Delta (x)| e^{i \phi (x)}$ with {\it nonzero gap modulus} $|\Delta|$ but random phase $\phi(x)$ so there is no phase coherence (or no quasi-long-range order in the case of a purely 2D system) and system behaves like a gas of non-condensed composite bosons. In superconductivity this region calls {\it pseudogap phase}. In this paper we rederive these results known from superconductivity in a very simple and trasparent way in a GN model with U(1)-symmetry and discuss its possible relations to particle physics. In particular we discuss possibility of generalization of these results to higher dimensions and other symmetries and roles of quantum and classical fluctuations. In the regime of large $N$ we also show that the temperature of the phase transition of XY-model tends from below to the temperature of the pair formation and merges with it in the limit $N \to \infty$ thus recovering mean-field scenario for the onset of quasi-long-range order in this model.