Relevance of recursive lattice approximations for description of frustrated magnetic systems: Star kagome-like recursive lattice approximation

We introduce and investigate in detail thermodynamic properties of the antiferromagnetic spin-1∕2 Ising model with the multisite interaction in the presence of the external magnetic field on the star kagome-like recursive lattice that takes into account not only basic triangular structure of the regular kagome lattice responsible for the geometric frustration but also a typical star-like structure of the kagome lattice consisting of six site-connected triangles. It is shown that the model exhibits single solution for arbitrary values of the parameters of the model, which is driven by a sixth-order polynomial equation. The free energy per site of the model is found and the magnetization, the entropy, and the specific heat capacity properties of the model are investigated and compared to those obtained in the framework of the model on the kagome-like Husimi lattice. The system of all ground states of the model is determined and exact expressions for their magnetization properties as well as for their residual entropies are derived. It is shown that all results obtained in the framework of the model on the star kagome-like recursive lattice are unprecedentedly close to those obtained on the much simpler kagome-like Husimi lattice. Besides, in the case when an exact result of the model on the real kagome lattice exists it is very close to the corresponding results obtained on the recursive lattices. These quite surprising but nontrivial facts allow one to make important assumptions about the properties of the model on the real two-dimensional kagome lattice even in the nonzero external magnetic field. First of all, one can suppose that various thermodynamic properties of the model on the kagome lattice must be qualitatively the same as the corresponding properties obtained in the framework of the model on various kagome-like recursive lattices. Moreover, one can also suppose that even the quantitative results obtained for the model on various kagome-like recursive lattices must be very close (it seems that some of them are even identical) to those that should be valid on the regular kagome lattice, although the exact solution of the model on the kagome lattice does not exist yet.