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The spin-$1/2$ model system with antiferromagnetic (AF) couplings on a$J_1$-$J_2$ checkerboard lattice, known as the planar pyrochlore model, isstrongly 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 bondsolid (VBS) state to be the ground state at $J_1\lesssim J_2$, while 1D AFcorrelated states take over the rest. We verify the stability of the VBS stateagainst nematic perturbation. The corresponding bosonic picture provides anintuitive understanding of the low-energy physics. Particularly, it predicts aweaker VBS states in the easy-plane limit, which we confirm numerically. Ourresults clarify the most essential question of this interesting system, anddemonstrate the usefulness of bosonic picture in dealing with frustratedmagnetism.

Nearly degenerate ground states of a checkerboard antiferromagnet and their bosonic interpretation