Classical hydrodynamics is a remarkably versatile description of the coarse-grained behaviour of many-particle systems once local equilibrium has been established^{1}. 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^{2,3}. 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^{4,5}. 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.

https://www.nature.com/articles/s41567-021-01191-6?proof=t%C2%A0

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