Symmetries, Anomalies, and Phases of the Chiral Gross-Neveu Model

The chiral Gross-Neveu model, in the large N limit and at finite temperature and density, displays spontaneous symmetry breaking of its axial U (1) symmetry and of space translations to a diagonal subgroup, signalled by a complex condensate periodic in space - dubbed chiral spiral. In this thesis we analytically investigate the possibility of remnants of such condensate, both at zero and finite temperature, in the finite N setup. Using the tool of non-Abelian bosonization, with a careful treatment of certain global aspects, we are able to relate the model to a SU (N ) × U (1) Wess-Zumino-Witten model with appropriate levels. At T = 0, we are able to rigorously prove that the chiral spiral configuration seen at large N persists at any finite N ≥ 2. At finite T, instead, we study the two-point function of certain composite fermion operators and we are able to predict that at finite temperature spatially modulated structures still exist, provided that one puts periodic boundary conditions for fermions in the thermal cycle. This distinction is motivated by the existence of an ’t Hooft anomaly for a D8 symmetry in the fermionic theory. In support of the finite N results we also rederive the large N chiral Gross-Neveu model phase diagram, assuming different boundary conditions for fermions, with a direct diagrammatic computation.