  Renormalization group analysis for multi-band many-electron systems at
half-filling at positive temperature is presented. The analysis includes the
Matsubara ultra-violet integration and the infrared integration around the zero
set of the dispersion relation. The multi-scale integration schemes are
implemented in a finite-dimensional Grassmann algebra indexed by discrete
position-time variables. In order that the multi-scale integrations are
justified inductively, various scale-dependent estimates on Grassmann
polynomials are established. We apply these theories in practice to prove that
for the half-filled Hubbard model with nearest-neighbor hopping on a square
lattice the infinite-volume, zero-temperature limit of the free energy density
exists as an analytic function of the coupling constant in a neighborhood of
the origin if the system contains the magnetic flux pi (mod 2 pi) per plaquette
and 0 (mod 2 pi) through the large circles around the periodic lattice.
Combined with Lieb's result on the flux phase problem ([Lieb, E. H., Phys. Rev.
Lett. 73 (1994), 2158]), this theorem implies that the minimum free energy
density of the flux phase problem converges to an analytic function of the
coupling constant in the infinite-volume, zero-temperature limit. The proof of
the theorem is based on a four-band formulation of the model Hamiltonian and an
extension of Giuliani-Mastropietro's renormalization designed for the
half-filled Hubbard model on the honeycomb lattice ([Giuliani, A. and V.
Mastropietro, Commun. Math. Phys. 293 (2010), 301-346]).
