The search for topological insulators has been actively promoted in the field of condensed matter physics for further development in energy-efficient information transmission and processing. In this context, recent studies have revealed that not only electrons but also bosonic particles such as magnons can construct edge states carrying nontrivial topological invariants. Here we demonstrate topological triplon bands in the spin-1/2 two-dimensional dimerized quantum antiferromagnet Ba$_2$CuSi$_2$O$_6$Cl$_2$, which is closely related to a pseudo-one-dimensional variant of the Su-Schrieffer-Heeger (SSH) model, through inelastic neutron scattering experiments. The excitation spectrum exhibits two triplon bands and a clear band gap between them due to a small alternation in interdimer exchange interactions along the $a$-direction, which is consistent with the crystal structure. The presence of topologically protected edge states is indicated by a bipartite nature of the lattice.
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