Thermodynamic Properties of Ising Spins on the Triangular Kagom\'{e} Lattice

The triangular Kagom\'{e} lattice can be constructed by inserting a lattice site on each bond of the Kagom\'{e} lattice. Each unit cell contains 6 $a$-sites, 3 $b-$sites, 6 \textit{aa-}bonds and 12 \textit{ab-}bonds. The lattice with antiferromagnetic \textit{aa}-bonds is known to exhibit geometrical frustration at low temperatures. We applied analytical methods and Monte Carlo simulations to study a system of Ising Spins on the lattice and investigated its thermodynamics properties. In particular, the heat capacity of the model exhibits interesting features based on the strength and the sign of coupling constants J$_{aa}$ and J$_{ab}$. In the case when the \textit{aa-}interaction is antiferromagnetic, the heat capacity shows two broad peaks at kT/$\vert $J$_{ab}\vert \cong $1.8 and at kT/$\vert $J$_{aa}\vert \cong $1.8. In the case when the \textit{aa-}interaction is ferromagnetic, the heat capacity shows a sharp peak at kT/$\vert $J$_{ab}\vert \cong $1.9 and another low broad peak at around kT/$\vert $J$_{aa}\vert \cong $1.4. We also studied a much simpler system of two $a-$trimers connected with $b-$sites to reproduce the thermodynamics behaviors of the more complicated triangular Kagom\'{e} lattice and to further understand the origin of its interesting properties.