  We study the high magnetic field regime of the antiferromagnetic insulator
Cs$_2$CuCl$_4$ by expressing the spin-1/2 operators in the relevant Heisenberg
model in terms of hard-core bosons and implementing the hard-core constraint
via an infinite on-site interaction. We focus on the case where the external
magnetic field exceeds the saturation field $B_{c}\approx8.5\;\mathrm{T}$ and
is oriented along the crystallographic $a$ axis perpendicular to the lattice
plane. Because in this case the excited states are separated by an energy gap
from the ground state, we may use the self-consistent ladder approximation to
take the strong correlations due to the hard-core constraint into account. In
Cs$_2$CuCl$_4$ there are additional interactions besides the hard-core
interaction which we treat in self-consistent Hartree-Fock approximation. We
calculate the spectral function of the hard-core bosons from which we obtain
the in-plane components of the dynamic structure factor, the magnetic
susceptibility, and the specific heat. Our results for the specific heat are in
good agreement with the available experimental data. We conclude that the
self-consistent ladder approximation in combination with a self-consistent
Hartree-Fock decoupling of the non-hard-core interactions gives an accurate
description of the physical properties of gapped hard-core bosons in two
dimensions at finite temperatures.
