XXZ and Ising spins on the triangular kagome lattice

The recently fabricated two-dimensional magnetic materials Cu9X2 cpa 6 ·xH2O cpa=2-carboxypentonic acid and X=F,Cl,Br have copper sites which form a triangular kagome lattice TKL , formed by introducing small triangles “a-trimers” inside of each kagome triangle “b-trimer” . We show that in the limit where spins residing on b-trimers have Ising character, quantum fluctuations of XXZ spins residing on the a-trimers can be exactly accounted for in the absence of applied field. This is accomplished through a mapping to the kagome Ising model, for which exact analytic solutions exist. We derive the complete finite-temperature phase diagram for this XXZ-Ising model, including the residual zero-temperature entropies of the seven ground-state phases. Whereas the disordered spin liquid ground state of the pure Ising TKL model has macroscopic residual entropy ln 72=4.2767. . . per unit cell, the introduction of transverse quantum couplings between neighboring a-spins reduces this entropy to 2.5258. . . per unit cell. In the presence of applied magnetic field, we map the TKL XXZ-Ising model to the kagome Ising model with three-spin interactions and derive the ground-state phase diagram. A small or even infinitesimal field leads to a new phase that corresponds to a nonintersecting loop gas on the kagome lattice, with entropy 1.4053. . . per unit cell and a mean magnetization for the b-spins of 0.12 1 per site. In addition, we find that for moderate applied field, there is a critical spin liquid phase that maps to close-packed dimers on the honeycomb lattice, which survives even when the a-spins are in the Heisenberg limit.