Breakdown of an Electric-Field Driven System: a Mapping to a Quantum   Walk
  Quantum transport properties of electron systems driven by strong electric fields are studied by mapping the Landau-Zener transition dynamics to a quantum walk on a semi-infinite one-dimensional lattice with a reflecting boundary, where the sites correspond to energy levels and the boundary the ground state. Quantum interference induces a distribution localized around the ground state, and when the electric field is strengthened, a delocalization transition occurs describing breakdown of the original electron system. 
Takashi Oka, Norio Konno, Ryotaro Arita and Hideo Aoki