  We propose a realistic scheme to implement discrete-time quantum walks in the
Brillouin zone (i.e., in quasimomentum space) with a spinor Bose-Einstein
condensate. Relying on a static optical lattice to suppress tunneling in real
space, the condensate is displaced in quasimomentum space in discrete steps
conditioned upon the internal state of the atoms, while short pulses
periodically couple the internal states. We show that tunable twisted boundary
conditions can be implemented in a fully natural way by exploiting the
periodicity of the Brillouin zone. The proposed setup does not suffer from
off-resonant scattering of photons and could allow a robust implementation of
quantum walks with several tens of steps at least. In addition, onsite
atom-atom interactions can be used to simulate interactions with infinitely
long range in the Brillouin zone.
