Thermodynamics of Ising Spins on the Triangular Kagome Lattice*

In the compounds $\mbox{Cu}_{9}\mbox{X}_2(\mbox{cpa})_{6}\cdot x\mbox{H}_2\mbox{O}$ (cpa=2-carboxypentonic acid; X=F,Cl,Br), the Cu spins form a fascinating and unique pattern called a triangular kagome lattice (TKL). We present a detailed study of Ising spins on such a lattice using exact methods and Monte Carlo simulation. We calculate the free energy, internal energy, specific heat, entropy, sublattice magnetizations, and susceptibility, and we find a rich phase diagram as a function of coupling constants, temperature, and applied magnetic field. In the frustrated regime at $T=0$, the system effectively decouples into independent degrees of freedom, giving residual entropy $s_0=\frac{1}{9} \ln 72$ per spin and correlation length $\xi=0$ -- an interesting contrast with the triangular and kagome lattice Ising models. Applying a field induces a critical phase (related to the honeycomb lattice dimer model) that has irrational entropy $0.0359$ per spin and $1/r^2$ correlations that should be detectable by neutron scattering.