Condensates, crystals, and renormalons in the Gross-Neveu model at finite density

We study the $O(2N)$ symmetric Gross-Neveu model at finite density in the presence of a $U(1)$ chemical potential $h$ for a generic number $a\le N-2$ of fermion fields. By combining perturbative quantum field theory, semiclassical large $N$, and Bethe ansatz techniques, we show that at finite $N$ two new dynamically generated scales ${\mathrm{\Lambda }}_{\mathrm{n}}$ and ${\mathrm{\Lambda }}_{\mathrm{c}}$ appear in the theory, governing the mass gap of neutral and charged fermions, respectively. Above a certain threshold value for $h$, $a$-fermion bound states condense and form an inhomogeneous configuration, which at infinite $N$ is a crystal spontaneously breaking translations. At large $h$, this crystal has mean ${\mathrm{\Lambda }}_{\mathrm{n}}$ and spatial oscillations of amplitude $2{\mathrm{\Lambda }}_{\mathrm{c}}$. The two scales also control the nonperturbative corrections to the free energy, resolving a puzzle concerning fractional-power renormalons and predicting new ones.