  We analyze the out-of-equilibrium dynamics of a quantum particle coupled to
local magnetic degrees of freedom that undergo a classical phase transition.
Specifically, we consider a two-dimensional tight-binding model that interacts
with a background of classical spins in thermal equilibrium, which are subject
to Ising interactions and act as emergent, correlated disorder for the quantum
particle. Particular attention is devoted to temperatures close to the
ferromagnet-to-paramagnet transition. To capture the salient features of the
classical transition, namely the effects of long-range correlations, we focus
on the strong coupling limit, in which the model can be mapped onto a quantum
percolation problem on spin clusters generated by the Ising model. By
inspecting several dynamical probes such as energy level statistics, inverse
participation ratios, and wave-packet dynamics, we provide evidence that the
classical phase transition might induce a delocalization-localization
transition in the quantum system at certain energies. We also identify further
important features due to the presence of Ising correlations, such as the
suppression of compact localized eigenstates.
