What happens if we increase the quasimomentum in a crystal above the edge of the Brillouin zone? If we apply an electric field to a 1D lattice so that the quasimomentum increases as $$\langle\dot q\rangle=eE$$
what happens when we reach the limit $\frac \pi a$? Does the quasimomentum cycle round to $-\frac \pi a$ or does it jump to the higher Brillouin zone?
 A: It cannot jump to a higher Brillouin zone, since the Brillouin zones are just equivalent descriptions of the toroidal momentum space of a system with a discrete translation symmetry. In a way the choice of the Brillouin zone is a choice of coordinates. So if we keep using the same choice of coordinates then the momentum wraps around from $\pi/a$ to $-\pi/a$.
This effect is known as Bloch oscillation: The velocity of the electron (and therefore the position) oscillates. In other words in a perfect metal with infinite extent a constant voltage causes an alternating current.
Note, however, that a boundary of the metal scatters the electrons and thereby inhibits the effects. Further, electron-electron interactions and phonon-electron interactions, impurites, and, if these are eradicated, Frenkel pairs mean that it requires very low temperatures and high electric fields to reach the domain where Bloch oscillations actually occur, as soon as there are strong enough scattering effects, the electron momentum will not wrap around but scattering will limit the acquired momentum.
