The zero-temperature limit of the free energy density in many-electron systems at half-filling

Abstract

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.

159 pages. Some changes in Introduction, the statement of Lemma 3.4, the proofs of Lemma 4.9 (1), Lemma 4.10 and References. Remark 3.5 was newly written. The rest remain unchanged. To appear in J. Math. Sci. Univ. Tokyo