  A second order cross-linked network is applied onto the classical Husimi
lattice, to investigate the role of a "phantom" non-neighboring interactions of
mid- and long-range in Bethe-like lattices for the first time. Since
antiferromagnetic Ising model on Husimi lattice has been exactly solved and
successfully presented the melting and glass transition, the Phantom Cross-link
Network (PCN) is introduced here to understand the relationship between glassy
defect and long-range interactions in small molecule systems, and the concept
is inspired from the classical rubber network theory (Flory, 1985). One random
site out of four on the recursive unites with certain distance I (the net size)
is selected to be linked onto the PCN. The solutions are still in the fashion
of normal antiferromagnetic Ising model, with expected frustrations along with
the net size I. Beside the regular Curie transition, several interesting
thermodynamics are observed in this toy model, and as the main found, PCN
clearly introduces glassy portion into the system, identified by the
supercooling behavior with lower TC, the metastable entropy curve and the
Kauzmann paradox.
