Theory of directed localization in one dimension

We present an analytical solution of the delocalization transition that is induced by an imaginary vector potential in a disordered chain [N. Hatano and D. R. Nelson, Phys. Rev. Lett. 77, 570 (1996)]. We compute the relation between the real and imaginary parts of the energy in the thermodynamic Limit, as well as finite-size effects. The results are in good agreement with numerical simulations for weak disorder (in which the mean free path is large compared to the wavelength).