  I point out that the $U(N)$ Chern-Simons $3d$ theory coupled to fermions at
finite temperature and at a specific mean field approximation and the $3d$
Gross-Neveu model at finite temperature and imaginary chemical potential can
give us the same results for the thermodynamic values of the free-energy and
the saddle point equation for the thermal mass. I further argue that the
periodic structure of the imaginary chemical potential brings also Bloch's
theorem into the game. Namely, the vacuum structure of the fermionic system
with imaginary baryon density is a Bloch wave. I further emphasise that Bloch
waves correspond to fermionic (antisymmetric) or bosonic (symmetric)
quasi-particles depending on the point in the band one sits in.
