  We prove by means of a renormalization group method that in weakly
interacting many-electron systems at half-filling on a periodic hyper-cubic
lattice, the free energy density uniformly converges to an analytic function of
the coupling constants in the infinite-volume, zero-temperature limit if the
external magnetic field has a chessboard-like flux configuration. The spatial
dimension is allowed to be any number larger than 1. The system covers the
Hubbard model with a nearest-neighbor hopping term, on-site interactions,
exponentially decaying density-density interactions and exponentially decaying
spin-spin interactions. The magnetic field must be included in the kinetic term
by the Peierls substitution. The flux configuration and the sign of the
nearest-neighbor density-density/spin-spin interactions can be adjusted so that
the free energy density is minimum among all the flux configurations.
Consequently, the minimum free energy density is proved to converge to an
analytic function of the coupling constants in the infinite-volume,
zero-temperature limit. These are extension of the results on a square lattice
in the preceding work ([Kashima, Y., "The special issue for the 20th
anniversary", J. Math. Sci. Univ. Tokyo. 23 (2016), 1-288]). We refer to lemmas
proved in the reference in order to complete the proof of the main results of
this paper. So this work is a continuation of the preceding work.
