The frustrated Heisenberg antiferromagnet on the checkerboard lattice: the $J_1$--$J_2$ model

We study the ground-state (gs) phases of the spin-half anisotropic planar pyrochlore (or crossed chain) model using the coupled cluster method (CCM). The model is a frustrated antiferromagnetic (AFM) $J_1$–$J_2$ system on the checkerboard lattice, with nearest-neighbor exchange bonds $J_1>0$ and next-nearest-neighbor bonds $J_2\equiv \kappa J_1>0$. Using various AFM classical ground states as CCM model states we present results for their gs energy, average on-site magnetization, and susceptibilities to plaquette valence-bond crystal (PVBC) and crossed-dimer valence-bond crystal (CDVBC) ordering. We show that the state with Neel ordering is the gs phase for $\kappa < \kappa_{c_1} \approx 0.80 \pm 0.01$, but that none of the fourfold set of AFM states selected by quantum fluctuations at $O(1\mathrm{/}s)$ in a large-$s$ analysis (where $s$ is the spin quantum number) from the infinitely degenerate set of AFM states that form the gs phase for the classical version of the model (for $\kappa >1$) survives the quantum fluctuations to form a stable magnetically-ordered gs phase for the spin-half case. The Neel state becomes susceptible to PVBC ordering at or very near to $\kappa ={\kappa }_{c_1}$, and the fourfold AFM states become infinitely susceptible to PVBC ordering at $\kappa ={\kappa }_{c_2}\approx 1.22\pm 0.02$. In turn, we find that these states become infinitely susceptible to CDVBC ordering for all values of $\kappa$ above a certain critical value at or very near to $\kappa = \kappa_{c_2}$\mathrm{.}WethusfindaNeel-orderedgsphasefor$\kappa<\kappa_{c_1}$, a PVBC-ordered phase for ${\kappa }_{c_1}<\kappa <{\kappa }_{c_2}$, and a CDVBC-ordered phase for $\kappa >{\kappa }_{c_2}$. Both transitions are probably direct ones, although we cannot exclude very narrow coexistence regions confined to $0.79\lesssim \kappa \lesssim 0.81$ and $1.20 \lesssim \kappa \lesssim 1.22$ respectively.