Frustration-free free fermions

We develop a general theory of frustration-free free-fermion systems and derive their necessary and sufficient conditions. Assuming locality and translation invariance, we find that the possible band touching between the valence bands and the conduction bands is always quadratic or softer, which rules out the possibility of describing Dirac and Weyl semimetals using frustration-free local Hamiltonians. We further construct a frustration-free free-fermion model on the honeycomb lattice and show that its density fluctuations acquire an anomalous gap originating from the diverging quantum metric associated with the quadratic band-touching points. Nevertheless, an $O(1/L^2)$ finite-size scaling of the charge-neutral excitation gap can be verified even in the presence of interactions, consistent with the more general results we derive in an accompany work [arXiv:2503.12879].