  The two-dimensional $S=1/2$ asymmetric Heisenberg Mattis model is
investigated with the exact diagonalization of finite clusters. The N\'eel
order parameter and the spin glass order parameter can be smoothly extrapolated
to the thermodynamic limit in the antiferromagnetic region, as in the pure
Heisenberg antiferromagnet. The critical concentration of the N\'eel phase is
consistent with that of the two-dimensional Ising Mattis model, and the spin
glass order parameter increases monotonously as the ferro-bond concentration
increases. These facts suggest that quantum fluctuation does not play an
essential role in two-dimensional non-frustrated random spin systems.
  KEYWORDS: quantum spin system, ground state, randomness, Mattis model, N\'eel
order, spin glass order
