  Classical electromagnetism is linear. However, fields can polarize the vacuum
Dirac sea, causing quantum nonlinear electromagnetic phenomena, e.g.,
scattering and splitting of photons that occur only in very strong fields found
in neutron stars or heavy ion colliders. We show that strong nonlinearity
arises in Dirac materials at much lower fields $\sim 1\:\text{T}$, allowing us
to explore the extremely high field limit of quantum electrodynamics in solids.
We explain recent experiments in a unified framework and predict nonlinear
magneto-electric response, including a magnetic enhancement of dielectric
constant and electrically induced magnetization. We propose experiments and
discuss the applications on novel materials.
