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We have numerically studied the thermodynamic properties of the spin $\frac{1}{2}$ 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)].
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