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In contrast to electron (fermion) systems, topological phases of chargeneutral bosons have been poorly understood despite recent extensive research oninsulating magnets. The most important unresolved issue is how the inevitableinter-bosonic interactions influence the topological properties. It has beenproposed that the quantum magnet SrCu$_2$(BO$_3$)$_2$ with an exact groundstate serves as an ideal platform for this investigation, as the system isexpected to be a magnetic analogue of a Chern insulator with electrons replacedby bosonic magnetic excitations (triplons). Here, in order to examinetopologically protected triplon chiral edge modes in SrCu$_2$(BO$_3$)$_2$, wemeasured and calculated the thermal Hall conductivity $\kappa_{xy}$. Ourcalculations show that the sign of $\kappa_{xy}$ is negative, which is oppositeto the previous calculations, and its magnitude is 2$\pi$ times smaller. Nodiscernible $\kappa_{xy}$ was observed, which is at most 20-30% of ourcalculations if present. This implies that even relatively weak inter-particleinteractions seriously influence the topological transport properties at finitetemperatures. These demonstrate that, in contrast to fermionic cases, thepicture of non-interacting topological quasi-particles cannot be naivelyapplied to bosonic systems, calling special attention to the interpretation ofthe topological bosonic excitations reported for various insulating magnets.

Intrinsic suppression of topological thermal Hall effect in an exactly solvable quantum magnet