  Finding an exact solution for a realistic interacting quantum many-body
problem is often challenging. There are only a few problems where an exact
solution can be found, usually in a narrow parameter space. Here, we propose a
spin-$1/2$ Heisenberg model on a square lattice with spatial anisotropy and
bond depletion for the nearest-neighbor antiferromagnetic interactions but not
for the next-nearest-neighbor interactions. This model has an \emph{exact} and
\emph{unique} dimer ground state at $J_2/J_1=1/2$; a dimer state is a product
state of spin-singlets on dimers (here, staggered nearest-neighbor bonds). We
examine this model by employing the bond-operator mean-field theory and exact
diagonalization. These analytical and numerical methods precisely affirm the
correctness of the dimer ground state at the exact point ($J_2/J_1=1/2$). As
one moves away from the exact point, the dimer order melts and vanishes when
the spin gap becomes zero. The mean-field theory with harmonic approximation
indicates that the dimer order persists for $-0.35\lesssim J_2/J_1\lesssim
1.35$. However, in non-harmonic approximation, the upper critical point lowers
by $0.28$ to $1.07$, but the lower critical point remains intact. The exact
diagonalization results suggest that the latter approximation fares better. The
model reveals N\'eel order below the lower critical point and stripe magnetic
order above the upper critical point. It has a topologically equivalent model
on a honeycomb lattice where the nearest-neighbor interactions are still
spatial anisotropic, but the bond depletion shifts into the isotropic
next-neighbor interactions. Moreover, these models can also be generalized in
the three dimensions.
