  In cuprate superconductors, superconductivity is consistently accompanied by
antiferromagnetism. This raises the question of a potential causal link between
superconductivity and antiferromagnetic mechanisms. In this study, we consider
the non-local Coulomb interaction and solve the single-particle Green function
exactly. The exact solution shows the existence of strong inter-spin
correlations in the absence of superexchange, eliminating the directional
degrees of freedom for spin quantization in the Hatsugai-Kohmoto (HK) model.
Strong Coulomb interactions force the energy band splitting into infinitely
equidistant sub-bands weighted by a Poisson distribution, exhibiting boson
properties (incoherent pairs of fermions) and non-Fermi liquids. Attractive
pairing interactions caused by electron-phonon coupling open a pairing gap at
the Fermi surface. Based on the Pauli exclusion principle, paired particles
with the same spin quantization direction completely occupy the state below the
gap, resulting in pairing insulators, superconducting instability, and
especially antiferromagnetism. A pseudogap occurs when the higher split
sub-band overlaps in energy with the lowest sub-band. The critical temperature
Tc of superconductivity exhibits a multi-dome structure and peaks around the
maximum density of states. The maximum Tc depends on splitting sub-bands and
particle occupancy, consistent with experiments. This study concludes that the
non-local effect preserves the Hubbard and HK models' characteristics while
renormalizing the system to antiferromagnetism in pairing insulators and Pauli
paramagnetism in weak semimetals.
