Revisiting the spatially inhomogeneous condensates in the $(1 + 1)$-dimensional chiral Gross-Neveu model via the bosonic two-point function in the infinite-$N$ limit

This work shows that the known phase boundary between the phase with chiral symmetry and the phase of spatially inhomogeneous chiral symmetry breaking in the phase diagram of the $(1 + 1)$-dimensional chiral Gross-Neveu model can be detected from the bosonic two-point function alone and thereby confirms and extends previous results arXiv:hep-th/0008175, arXiv:0807.2571, arXiv:0909.3714, arXiv:1810.03921, arXiv:2203.08503. The analysis is referred to as the stability analysis of the symmetric phase and does not require knowledge about spatial modulations of condensates. We perform this analysis in the infinite-$N$ limit at nonzero temperature and nonzero quark and chiral chemical potentials also inside the inhomogeneous phase. Thereby we observe an interesting relation between the bosonic $1$-particle irreducible two-point vertex function of the chiral Gross-Neveu model and the spinodal line of the Gross-Neveu model.