Inhomogeneities in the \(2\)-Flavor Chiral Gross-Neveu Model

We investigate the finite-temperature and -density chiral Gross-Neveu model with an axial U\(_A\)(1) symmetry in \(1+1\) dimensions on the lattice. In the limit where the number of flavors \(N_\mathrm{f}\) tends to infinity the continuum model has been solved analytically and shows two phases: a symmetric high-temperature phase with a 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 to \(N_\mathrm{f}\to\infty\), we find for \(8\) flavors, where quantum and thermal fluctuations are suppressed, two distinct regimes in the \((T,\mu)\) phase diagram, characterized by qualitatively different behavior of the two-point functions of the condensate fields. More surprisingly, at \(N_\mathrm{f}=2\), where fluctuations are no longer suppressed, the model still behaves similarly to the \(N_\mathrm{f}\to\infty\) model and we conclude that the chiral spiral leaves its footprints even on systems with a small number of flavors. For example, at low temperature the two-point functions are still dominated by chiral spirals with pitches proportional to the inverse chemical potential, although in contrast to large-\(N_\mathrm{f}\) their amplitudes decrease with distance. We argue that these results should not be interpreted as the spontaneous breaking of a continuous symmetry, which is forbidden in two dimensions. Finally, using 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.