A weak-value model for virtual particles supplying the electric current   in graphene: the minimal conductivity and the Schwinger mechanism
  We propose a model for the electric current in graphene in which electric carriers are supplied by virtual particles allowed by the uncertainty relations. The process to make a virtual particle real is described by a weak value of a group velocity: the velocity is requisite for the electric field to give the virtual particle the appropriate changes of both energy and momentum. With the weak value, we approximately estimate the electric current, considering the ballistic transport of the electric carriers. The current shows the quasi-Ohimic with the minimal conductivity of the order of e^2/h per channel. Crossing a certain ballistic time scale, it is brought to obey the Schwinger mechanism. 
Kazuhiro Yokota, Nobuyuki Imoto
quant-ph cond-mat.mes-hall