  We report high-resolution single-crystal inelastic neutron scattering
measurements on the spin-1/2 antiferromagnet Ba(TiO)Cu$_4$(PO$_4$)$_4$. This
material is formed from layers of four-site \cupola" structures, oriented
alternately upwards and downwards, which constitute a rather special
realization of two-dimensional (2D) square-lattice magnetism. The strong
Dzyaloshinskii-Moriya (DM) interaction within each cupola, or plaquette, unit
has a geometry largely unexplored among the numerous studies of magnetic
properties in 2D Heisenberg models with spin and spatial anisotropies. We have
measured the magnetic excitations at zero field and in fields up to 5 T,
finding a complex mode structure with multiple characteristic features that
allow us to extract all the relevant magnetic interactions by modelling within
the linear spin-wave approximation. We demonstrate that
Ba(TiO)Cu$_4$(PO$_4$)$_4$ is a checkerboard system with almost equal intra- and
inter-plaquette couplings, in which the intra-plaquette DM interaction is
instrumental both in enforcing robust magnetic order and in opening a large gap
at the Brillouin-zone center. We place our observations in the perspective of
generalized phase diagrams for spin-1/2 square-lattice models and materials,
where exploring anisotropies and frustration as routes to quantum disorder
remains a frontier research problem.
