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

Fermions coupled to Chern-Simons gauge field or imaginary chemical potential and the Bloch theorem