Symmetry breaking and competing valence bond states in the star lattice Heisenberg antiferromagnet

We investigate the ground state phase diagram of the spin-$1/2$ antiferromagnetic Heisenberg model on the star lattice using infinite projected entangled pair states (iPEPS) and high-order series expansions. The model includes two distinct couplings: ${J}_{d}$ on the dimer bonds and ${J}_{t}$ on the trimer bonds. While it is established that the system hosts a valence bond solid (VBS) phase for ${J}_{d}\ensuremath{\ge}{J}_{t}$, the ground state phase diagram for ${J}_{d}<{J}_{t}$ has remained unsettled. Our iPEPS simulations uncover a first-order phase transition at ${J}_{d}/{J}_{t}\ensuremath{\approx}0.18$, significantly lower than previously reported estimates. Beyond this transition, we identify a close competition between two valence bond crystal (VBC) states; a columnar VBC and a $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ VBC, with the latter consistently exhibiting lower energy across all finite bond dimensions. The high-order series expansion supports this by finding that the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ VBC state indeed becomes energetically favorable, but only at sixth order in perturbation theory, revealing the subtle nature of the competition between candidate states.