  We unveil the multifractal behavior of Ising spin glasses in their
low-temperature phase. Using the Janus II custom-built supercomputer, the
spin-glass correlation function is studied locally. Dramatic fluctuations are
found when pairs of sites at the same distance are compared. The scaling of
these fluctuations, as the spin-glass coherence length grows with time, is
characterized through the computation of the singularity spectrum and its
corresponding Legendre transform. A comparatively small number of site pairs
controls the average correlation that governs the response to a magnetic field.
We explain how this scenario of dramatic fluctuations (at length scales smaller
than the coherence length) can be reconciled with the smooth, self-averaging
behavior that has long been considered to describe spin-glass dynamics.

Disordered Systems and Neural Networks