  We have numerically studied the thermodynamic properties of the spin 1/2 XXZ
chain in the presence of a transverse (non commuting) magnetic field. The
thermal, field dependence of specific heat and correlation functions for chains
up to 20 sites have been calculated. The area where the specific heat decays
exponentially is considered as a measure of the energy gap. We have also
obtained the exchange interaction between chains in a bulk material using the
random phase approximation and derived the phase diagram of the three
dimensional material with this approximation. The behavior of the structure
factor at different momenta verifies the antiferromagnetic long range order in
y-direction for the three dimensional case. Moreover, we have concluded that
the Low Temperature Lanczos results [M. Aichhorn et al., Phys. Rev. B 67,
161103(R) (2003)] are more accurate for low temperatures and closer to the full
diagonalization ones than the results of Finite Temperature Lanczos Method [J.
Jaklic and P. Prelovsek, Phys. Rev. B 49, 5065 (1994)].
