Lieb-Schultz-Mattis type theorems for Majorana models with discrete symmetries

We prove two Lieb-Schultz-Mattis type theorems that apply to any translationally invariant and local fermionic $d$-dimensional lattice Hamiltonian for which fermion-number conservation is broken down the of fermion parity. show when internal symmetry group ${G}_{f}^{\phantom{\rule{0.16em}{0ex}}}$ realized locally (in a repeat unit cell lattice) by nontrivial projective representation, then ground state cannot be simultaneously nondegenerate, symmetric (with respect translations ${G}_{f}^{\phantom{\rule{0.16em}{0ex}}}$), gapped. also hosts an odd number Majorana degrees freedom cardinality even, gapped, translation symmetric.