Thermal critical dynamics from equilibrium quantum fluctuations

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.