  The conformal anomaly indicates the breaking of conformal symmetry
(angle-preserving transformations) in the quantum theory by quantum
fluctuations and is a close cousin of the gravitational anomaly. We show, for
the first time, that the conformal anomaly controls the variance of the local
magnetization $M_{loc}$ at finite temperatures in spin chains and spin ladders.
This effect is perceived at constant and variable temperature across the
sample. The change of $M_{loc}$ induced by the conformal anomaly is of the
order of 3-5\% of the maximal spin at one Kelvin for DIMPY or CuPzN and
increases linearly with temperature. Further, for a temperature gradient of
10\% across the sample, the time-relaxation of the non-equilibrium $M_{loc}$ is
of the order of nanoseconds. Thus, we believe that experimental techniques such
as neutron scattering, nuclear magnetic resonance~(NMR), spin noise and
ultrafast laser pumping should pinpoint the presence of the conformal anomaly.
Therefore, we pave the road to detect the conformal anomaly in spin observables
of strongly interacting low-dimensional magnets.
