  The Lieb-Schultz-Mattis (LSM) theorem and its higher-dimensional
generalizations by Oshikawa and Hastings establish that a translation-invariant
lattice model of spin-$1/2$'s can not have a non-degenerate ground state
preserving both spin and translation symmetries. Recently it was shown that LSM
theorems can be interpreted in terms of bulk-boundary correspondence of certain
weak symmetry-protected topological (SPT) phases. In this work we discuss
LSM-type theorems for two-dimensional fermionic systems, which have no bosonic
analogs. They follow from a general classification of weak SPT phases of
fermions in three dimensions. We further derive constraints on possible gapped
symmetry-enriched topological phases in such systems. In particular, we show
that lattice translations must permute anyons, thus leading to
"symmetry-enforced" non-Abelian dislocations, or "genons". We also discuss
surface states of other weak SPT phases of fermions.
