XXZand Ising spins on the triangular kagome lattice

Abstract

The recently fabricated two-dimensional magnetic materials ${\text{Cu}}_{9}{X}_{2}{(\text{cpa})}_{6}\ensuremath{\cdot}x{\text{H}}_{2}\text{O}$ ($\text{cpa}=2$-carboxypentonic acid and $X=\text{F},\text{Cl},\text{Br}$) 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 $\text{ln}\text{ }72=4.2767\dots{}$ per unit cell, the introduction of transverse (quantum) couplings between neighboring $a$-spins reduces this entropy to $2.5258\dots{}$ 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 $1.4053\dots{}$ 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.