Delocalization of a non-Hermitian quantum walk on random media in one dimension

Delocalization transition is numerically found in a non-Hermitian extension of discrete-time quantum walk on one-dimensional random medium. At the transition, an eigenvector gets delocalized and at same time corresponding energy eigenvalue (the imaginary unit times phase time-evolution operator) becomes complex. This accordance with Anderson model one dimension, called, Hatano-Nelson model. We thereby find that all eigenstates Hermitian share common localization length.