  The connection between symmetries and conservation laws is a cornerstone of
physics. It underlies Bloch's theorem which explains wave phenomena in all
linear periodic systems. Here we demonstrate that, in a nonlinear grating with
memory, diffracted waves can spontaneously acquire momentum parallel to the
lattice vector in quantities unconstrained by the grating period. In this
breakdown of Bloch's theorem, which we also evidence in solutions to nonlinear
Maxwell's equations, wave amplitudes no longer respect the discrete translation
symmetry of the grating. Our findings reveal a rich phenomenology for waves in
nonlinear periodic systems, and point to numerous opportunities for nonlinear
lattices with broken symmetry in the context of imaging, sensing, and
information processing in general.
