  We study the anisotropic quantum Heisenberg antiferromagnet for spin-1/2 that
interpolates smoothly between the one-dimensional (1D) and the two-dimensional
(2D) limits. Using the spin Hartree-Fock approach we construct a quantitative
theory of heat capacity in the quasi-1D regime with a finite coupling between
spin chains. This theory reproduces closely the exact result of Bethe Ansatz in
the 1D limit and does not produces any spurious phase transitions for any
anisotropy in the quasi-1D regime at finite temperatures in agreement with the
Mermin-Wagner theorem. We study the static spin-spin correlation function in
order to analyse the interplay of lattice geometry and anisotropy in these
systems. We compare the square and triangular lattice. For the latter we find
that there is a quantum transition point at an intermediate anisotropy of
$\sim0.6$. This quantum phase transition establishes that the quasi-1D regime
extends upto a particular point in this geometry. For the square lattice the
change from the 1D to 2D occurs smoothly as a function of anisotropy, i.e. it
is of the crossover type. Comparing the newly developed theory to the available
experimental data on the heat capacity of $\rm{Cs}_2\rm{CuBr}_4$ and
$\rm{Cs}_2\rm{CuCl}_4$ we extract the microscopic constants of the exchange
interaction that previously could only be measured using inelastic neutron
scattering in high magnetic fields.
