Understanding strongly interacting matter at finite density, such as QCD, remains a major challenge.
Insight can be gained from simpler theories like the $O(2N)$ symmetric 2d Gross-Neveu (GN) model. The GN model with chemical potentials has been studied for decades using diverse techniques and motivations: from exact scale-mass relations to inhomogeneous phases and, more recently, resurgence and renormalons.
We review these developments and unify them into a coherent framework for arbitrary $U(1)$ chemical potentials $h$. At finite $N$, two nonperturbative scales $\Lambda_\mathrm{n}$ and $\Lambda_\mathrm{c}$ emerge, governing neutral and charged fermion masses.  At large $N$, they also control the mean and the oscillations of an inhomogeneous crystal phase. 
These scales also control the leading nonperturbative corrections to the free energy, resolving a recent puzzle involving fractional-power renormalons and predicting new ones.