Spatially oscillating correlation functions in $\left(2+1\right)$-dimensional four-fermion models: The mixing of scalar and vector modes at finite density

In this work, we demonstrate that the mixing of scalar and vector condensates produces spatially oscillating, but exponentially damped correlation functions in fermionic theories at finite density and temperature. We find a regime exhibiting this oscillatory behavior in a Gross-Neveu-type model that also features vector interactions within the mean-field approximation. The existence of this regime aligns with expectations based on symmetry arguments, that are also applicable to QCD at finite baryon density. We compute the phase diagram including both homogeneous phases and regions with spatially oscillating, exponentially damped correlation functions at finite temperature and chemical potential for different strengths of the vector coupling. Furthermore, we find that inhomogeneous condensates are disfavored compared to homogeneous ones akin to previous findings without vector interactions. We show that our results are valid for a broad class of $\left(2+1\right)$-dimensional models with local four-fermion interactions.