Mass generation without phase coherence in Gross-Neveu Model with U(1) symmetry at finite temperature and low N

We show that the Gross-Neveu model with U(1) symmetry for a small number $N$ of fermions has two characteristic temperatures: $T_c$ that corresponds to onset of long-range or quasi-long-range order and $T^* (>T_c)$ that corresponds to thermal fermion pairs breaking. We show that the region $T_c<T<T^*$ is characterized 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 and system behaves like a gas of non-condesed composite bosons. Condensation (or quasicondensation, in the case of purely two-dimensional system) of preformed pairs in the regime of low $N$ can not be described by mean-field theory so we extract lowest gradient terms and set up an effective XY-model with temperature dependent stiffness coeficent that serves for description of the onset of the phase coherence in the system with preformed gap modulus(or description of the Kosterlitz-Thouless transition in a purely 2D system). In the regime of large $N$ we 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 long-range (or quasi-long-range) order in this model.