Phase diagram of the S=1/2 checkerboard antiferromagnet

We report the phase diagram of the S=1/2 Heisenberg antiferromagnet on the checkerboard lattice, also known as the crossed-chains lattice. It is assumed that the exchange coupling along the crossing chains, $J_2$, is different from that on the inter-chain (square-lattice) links, $J_1$. We show that in the one-dimensional limit $J_2 \gg J_1$, the ground state is spontaneously dimerized in the crossed-dimer pattern. At the isotropic planar pyrochlore point $J_2=J_1$, the ground state is also a valence-bond solid (VBS) but of the plaquette type, as is known from previous studies. We argue that these two VBS states may be connected, as a function of the ratio $J_2/J_1$, by an intermediate magnetically-ordered phase. Two quantum critical points, separating the ordered phase from dimerized VBS ones, are analyzed and argued to belong to the $O(3)$ universality class.