Thermodynamics of Ising spins on the triangular kagome lattice: Exact analytical method and Monte Carlo simulations

Abstract

A new class of two-dimensional magnetic materials ${\mathrm{Cu}}_{9}{X}_{2}{(\mathrm{cpa})}_{6}\ensuremath{\cdot}x{\mathrm{H}}_{2}\mathrm{O}$ ($\mathrm{cpa}=2$-carboxypentonic acid; $X=\mathrm{F},\mathrm{Cl},\mathrm{Br}$) was recently fabricated in which Cu sites form a triangular kagome lattice (TKL). As the simplest model of geometric frustration in such a system, we study the thermodynamics of Ising spins on the TKL using exact analytic method as well as Monte Carlo simulations. We present the free energy, internal energy, specific heat, entropy, sublattice magnetizations, and susceptibility. We describe the rich phase diagram of the model as a function of coupling constants, temperature, and applied magnetic field. For frustrated interactions in the absence of applied field, the ground state is a spin liquid phase with residual entropy per spin ${s}_{0}∕{k}_{B}=\frac{1}{9}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}72\ensuremath{\approx}0.4752\dots{}$ . In weak applied field, the system maps to the dimer model on a honeycomb lattice, with residual entropy 0.0359 per spin and quasi-long-range order with power-law spin-spin correlations that should be detectable by neutron scattering. The power-law correlations become exponential at finite temperatures, but the correlation length may still be long.