  $5d$ iridium oxides are of huge interest due to the potential for new quantum
states driven by strong spin-orbit coupling. The strontium iridate
Sr$_2$IrO$_4$ is particularly in the spotlight because of the so-called
$j_\text{eff}=1/2$ state consisting of a quantum superposition of the three
local $t_{2g}$ orbitals with -- in its most simple version -- nearly equal
population, which stabilizes an unconventional Mott insulating state. Here, we
report an anisotropic and aspherical magnetization density distribution
measured by polarized neutron diffraction in a magnetic field up to 5~T at 4~K,
which strongly deviates from a local \jeffHalf picture even when
distortion-induced deviations from the equal weights of the orbital populations
are taken into account. Once reconstructed by the maximum entropy method and
multipole expansion model refinement, the magnetization density shows
cross-shaped positive four lobes along the crystallographic tetragonal axes
with a large spatial extent, showing that the $xy$ orbital contribution is
dominant. The analogy to the superconducting copper oxide systems might then be
weaker than commonly thought.
