Fig. 1: Energy bands.
We consider translation-invariant free-fermionic models on a one-dimensional lattice, where the energy bands εj(k) are piecewise continuous functions of the wave number k and at least one of the energy bands is critical, that is, exhibits a non-trivial Fermi surface, here illustrated in solid green. The energy bands depicted in dashed magenta do not count as critical in our sense because the subsets of wave numbers k for which they are not strictly positive or not strictly negative, respectively, consist of only finitely many points: the bands do not cross the Fermi energy at 0.
