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.

Main text: 12 pages plus 5 figures. Supplemental Information: pages 18-36