  We used the Cartan formalism to construct fermionic models that are
compatible with Galilean or Carrollian symmetry and rigid scaling symmetry. The
free Carrollian fermion model exhibits conformal Carrollian symmetry which is
isomorphic to the asymptotic symmetries for flat spacetime in three dimensions,
namely the BMS$_3$ symmetry. We performed canonical quantization to this free
BMS fermion model, discussed both the highest weight vacuum and the induced
vacuum, calculated the correlation functions and the torus partition function.
Finally we constructed $\mathcal N=2$ supersymmetric theories by combining the
free fermion model and the free scalar model.
