  The spin-$1/2$ model system with antiferromagnetic (AF) couplings on a
$J_1$-$J_2$ checkerboard lattice, known as the planar pyrochlore model, is
strongly frustrated and associate with a two-to-one dimensional (2D-to-1D)
crossover. Using the Projected Entangled Simplex States tensor network ansatz,
we identify several nearly degenerate states in the frustrated region
($J_1<J_2$). Specifically, we find the long-sought crossed-dimer valence bond
solid (VBS) state to be the ground state at $J_1\lesssim J_2$, while 1D AF
correlated states take over the rest. We verify the stability of the VBS state
against nematic perturbation. The corresponding bosonic picture provides an
intuitive understanding of the low-energy physics. Particularly, it predicts a
weaker VBS states in the easy-plane limit, which we confirm numerically. Our
results clarify the most essential question of this interesting system, and
demonstrate the usefulness of bosonic picture in dealing with frustrated
magnetism.
