  Classical hydrodynamics is a remarkably versatile description of the
coarse-grained behavior of many-particle systems once local equilibrium has
been established. The form of the hydrodynamical equations is determined
primarily by the conserved quantities present in a system. Some quantum spin
chains are known to possess, even in the simplest cases, a greatly expanded set
of conservation laws, and recent work suggests that these laws strongly modify
collective spin dynamics even at high temperature. Here, by probing the
dynamical exponent of the one-dimensional Heisenberg antiferromagnet KCuF$_3$
with neutron scattering, we find evidence that the spin dynamics are well
described by the dynamical exponent $z=3/2$, which is consistent with the
recent theoretical conjecture that the dynamics of this quantum system are
described by the Kardar-Parisi-Zhang universality class. This observation shows
that low-energy inelastic neutron scattering at moderate temperatures can
reveal the details of emergent quantum fluid properties like those arising in
non-Fermi liquids in higher dimensions.
