We present a comprehensive study of magnon excitations in the tetragonal easy-plane anti-ferromagnet Bi$_2$CuO$_4$ using inelastic neutron scattering and spin wave analyses. The nature of low energy magnons, and hence the anisotropy in this material, has been controversial. We show unambiguously that the low energy magnon spectrum consists of a gapped and a gapless mode, which we attribute to out-of-plane and in-plane spin fluctuations, respectively. We modelled the observed magnon spectrum using linear spin wave analysis of a minimal anisotropic spin model motivated by the lattice symmetry. By studying the magnetic field dependence of the (1, 0, 0) Bragg peak intensity and the in-plane magnon intensity, we observed a spin-flop transition in the $ab$ plane at $sim0.4$~T which directly indicates the existence of a small in-plane anisotropy that is classically forbidden. It is only by taking into account magnon zero-point fluctuations beyond the linear spin wave approximation, we could explain this in-plane anisotropy and its magnitude, the latter of which is deduced from critical field of the spin-flop transition. The microscopic origins of the observed anisotropic interactions are also discussed. We found that our data is inconsistent with a large Dzyaloshinskii-Moriya interaction, which suggests a potential departure of Bi$_2$CuO$_4$ from the conventional theories of magnetic anisotropy for other cuprates.
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