  We study the quenched disordered magnetic system, which is obtained from the
2D SO(3) quantum Heisenberg model, on a square lattice, with nearest neighbors
interaction, by taking a Gaussian random distribution of couplings centered in
an antiferromagnetic coupling, $\bar J>0$ and with a width $\Delta J$. Using
coherent spin states we can integrate over the random variables and map the
system onto a field theory, which is a generalization of the SO(3) nonlinear
sigma model with different flavors corresponding to the replicas, coupling
parameter proportional to $\bar J$ and having a quartic spin interaction
proportional to the disorder ($\Delta J$). After deriving the CP$^1$ version of
the system, we perform a calculation of the free energy density in the limit of
zero replicas, which fully includes the quantum fluctuations of the CP$^1$
fields $z_i$. We, thereby obtain the phase diagram of the system in terms of
($T, \bar J, \Delta J$). This presents an ordered antiferromagnetic (AF) phase,
a paramagnetic (PM) phase and a spin-glass (SG) phase. A critical curve
separating the PM and SG phases ends at a quantum critical point located
between the AF and SG phases, at T=0. The Edwards-Anderson order parameter, as
well as the magnetic susceptibilities are explicitly obtained in each of the
three phases as a function of the three control parameters. The magnetic
susceptibilities show a Curie-type behavior at high temperatures and exhibit a
clear cusp, characteristic of the SG transition, at the transition line. The
thermodynamic stability of the phases is investigated by a careful analysis of
the Hessian matrix of the free energy. We show that all principal minors of the
Hessian are positive in the limit of zero replicas, implying in particular that
the SG phase is stable.
