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 $a \leq N-2$ of fermion fields. By combining perturbative QFT, semiclassical large $N$, and Bethe ansatz techniques, we show that at finite $N$ two new dynamically generated scales $\Lambda_\mathrm{n}$ and $\Lambda_\mathrm{c}$ 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 $\Lambda_\mathrm{n}$ and spatial oscillations of amplitude $2\Lambda_\mathrm{c}$. The two scales also control the nonperturbative corrections to the free energy, resolving a puzzle concerning fractional-power renormalons and predicting new ones.