  In contrast to electron (fermion) systems, topological phases of charge
neutral bosons have been poorly understood despite recent extensive research on
insulating magnets. The most important unresolved issue is how the inevitable
inter-bosonic interactions influence the topological properties. It has been
proposed that the quantum magnet SrCu$_2$(BO$_3$)$_2$ with an exact ground
state serves as an ideal platform for this investigation, as the system is
expected to be a magnetic analogue of a Chern insulator with electrons replaced
by bosonic magnetic excitations (triplons). Here, in order to examine
topologically protected triplon chiral edge modes in SrCu$_2$(BO$_3$)$_2$, we
measured and calculated the thermal Hall conductivity $\kappa_{xy}$. Our
calculations show that the sign of $\kappa_{xy}$ is negative, which is opposite
to the previous calculations, and its magnitude is 2$\pi$ times smaller. No
discernible $\kappa_{xy}$ was observed, which is at most 20-30% of our
calculations if present. This implies that even relatively weak inter-particle
interactions seriously influence the topological transport properties at finite
temperatures. These demonstrate that, in contrast to fermionic cases, the
picture of non-interacting topological quasi-particles cannot be naively
applied to bosonic systems, calling special attention to the interpretation of
the topological bosonic excitations reported for various insulating magnets.
