  We show that quantum fluctuations display a singularity at thermal critical
points, involving the dynamical $z$ exponent. Quantum fluctuations, captured by
the quantum variance (I. Fr\'erot and T. Roscilde, Phys. Rev. B 94, 075121
(2016)), can be expressed via purely static quantities; this in turn allows us
to extract the $z$ exponent related to the intrinsic Hamiltonian dynamics via
equilibrium unbiased numerical calculations, without invoking any effective
classical model for the critical dynamics. These findings illustrate that,
unlike classical systems, in quantum systems static and dynamic properties
remain inextricably linked even at finite-temperature transitions, provided
that one focuses on static quantities that do not bear any classical analog,
namely on quantum fluctuations.
