Inhomogeneities in the $2$-flavor Chiral Gross-Neveu model

We investigate the finite temperature and density chiral Gross-Neveu (cGN) model with axial U$_A$(1) symmetry in $1+1$ dimensions on the lattice. In the limit where the number of flavors $N_f$ tends to infinity the continuum model has been solved analytically and shows two phases: a symmetric high-temperature phase with vanishing condensate and a low-temperature phase in which the complex condensate forms a chiral spiral which breaks translation invariance. In the lattice simulations we employ chiral SLAC fermions with exact axial symmetry. Similarly as for $N_f\to\infty$ we find two distinct regimes in the $(T,\mu)$ phase diagram, characterized by a qualitatively different behavior of the two-point functions of the condensate fields. For $N_f=8$ flavors quantum and thermal fluctuations are suppressed and the cGN model behaves similarly as for $N_f\to\infty$. More surprisingly, at $N_f=2$, where fluctuations are no longer suppressed, the model still behaves similar to the $N_f\to\infty$ model and we conclude that the chiral spiral leaves its footprints even on the systems with a small number of flavors. For example, at low temperature the two-point functions are still dominated by chiral spirals with pitch proportional to the inverse chemical potential similarly as in the large-$N_f$ solution, although in contrast to large-$N_f$ the amplitude decreases with distance. With Dyson-Schwinger equations we calculate the decay of the U$_A$(1)-invariant fermion four-point function in search for a BKT phase at zero temperature.