Spin excitation of the Heisenberg antiferromagnet with frustration: from the bounce-lattice antiferromagnet through the maple-leaf-lattice antiferromagnet to the exact-dimer system

The spin-S Heisenberg antiferromagnet on the two-dimensional lattice is investigated for S=1/2 and S=1. We consider interaction at isolated dimers ($J_{\mathrm{d}}$) and interaction bonds that form the bounce lattice ($J_{\mathrm{b}}$). For $J_{\mathrm{d}}=J_{\mathrm{b}}$, the system is reduced to the maple-leaf-lattice antiferromagnet. We primarily conduct highly parallelized numerical diagonalization to examine the spin excitation gap above the ground state for various $J_{\mathrm{b}}\mathrm{/}{J}_{\mathrm{d}}$ cases. For S=1/2, we report calculations for a 42-site cluster that has not been previously treated. The S=1 case is examined for the first time for clusters up to 24 sites. Regardless of whether S=1/2 or 1, we find that the system has a gapped nature for small $J_{\mathrm{d}}\mathrm{/}{J}_{\mathrm{b}}$ and becomes gapless at $J_{\mathrm{d}}\mathrm{/}{J}_{\mathrm{b}}\sim 1.4$. For S=1, we also find that another gapped region appears between the gapless case at $J_{\mathrm{d}}\mathrm{/}{J}_{\mathrm{b}}\sim 1.4$ and the boundary of the exact-dimer phase.