Thermodynamics of Gross Neveu Models

We present the detailed properties of the phase diagrams of massless (1+1)-dimensional Gross-Neveu and Nambu-Jona-Lasinio models with discrete and continuous chiral symmetry by studying the exact, inhomogeneous solutions of the functional gap equation. We show that in addition to spontaneous breaking of continuous chiral symmetry, the Nambu-Jona-Lasinio model also exhibits translational symmetry breaking at finite density below a critical temperature. The spatially inhomogeneous phase, the chiral spiral is a periodic spiral in the chiral plane with a constant charge density. The mathematical method we develop to study the inhomogeneous phases, allows one to reduce the functional gap equation for the inhomogeneous condensate, to a nonlinear Schrödinger equation, which is exactly soluble. The general solution is a crystalline array of kinks and anti-kinks and includes as special cases all previously known real and complex condensate solutions to the gap equation. Furthermore, the associated Dirac equation is also soluble with this inhomogeneous chiral condensate, and the exact spectral properties are derived. Analyzing the thermodynamic properties of these solutions, we show the stability of the chiral spiral phase against the more general twisted kink crystal solution of the gap equation. This situation should be contrasted with the Gross-Neveu model, which has a discrete chiral symmetry, and for which the phase diagram has a crystalline phase with a periodic kink crystal. Then we argue that the existence of the chiral spiral phase is ubiquitous in (1+1)-dimensional models exhibiting continuous chiral symmetry. We further develop a connection between the Ginzburg-Landau expansion of the thermodynamic grand potential and a well-known mathematical integrable hierarchy known as the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy.