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The anisotropic $s=\frac{1}{2}$ anti-ferromagnetic Heisenberg chain in the
presence of an external magnetic field is studied by using the standard
quantum renormalization group. We obtain the critical line of the transition
from partially magnetized (PM) phase to the saturated ferromagnetic (SFM)
phase. The crossover exponent between the PM phase and anti-ferromagnetic
Ising (AFI) phase is evaluated. Our results show that the anisotropy
$(\Delta)$ term is relevant and causes crossover. These results indicate
that the standard RG approach yiels fairly good values for critical
points and their exponents. The magnetization
curve, correlation functions and the ground
state energy per site are obtained and compared with the known exact results.
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