Inversion symmetry breaking and criticality in free fermionic lattices

Abstract

We describe the connection between inversion symmetry breaking and criticality in free fermionic lattice models. It is shown that for translation-invariant spinless fermions, the breaking of this symmetry in the ground state implies criticality, i.e., the existence of long-range correlations and the vanishing of the spectral gap; while for models with spin, only the asymmetry of the spin-averaged covariance matrix implies a similar conclusion. Our results are proved by introducing invariants under global translation-invariant free fermion quenches. Using this result, we identify a set of models where the generalized Hartree-Fock approximation must break down.