Inhomogeneities in the $2$-Flavor Chiral Gross-Neveu Model

Abstract

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.

Comments: 15 pages + 5 pages appendix, 16 figures; added appendix on autocorrelations, added ancillary files, added clarifications