Keywords
Abstract
Localization of electrons in 1D disordered systems is usually described in the random phase approximation, when distributions of phases j and θ, entering the transfer matrix, are considered as uniform. In the general case, the random phase approximation is violated, and the evolution equations (when the system length L is increased) contain three independent variables, i.e. the Landauer resistance ρ and the combined phases ψ = θ - j and χ = θ + j. The phase χ does not affect the evolution of ρ and was not considered in previous papers. The distribution of the phase ψ is found to exhibit an unusual phase transition at the point Ɛ0 when changing the electron energy Ɛ, which manifests itself in the appearance of the imaginary part of ψ. The resistance distribution P(ρ) has no singularity at the point Ɛ0, and the transition looks unobservable in the electron disordered systems. However, the theory of 1D localization is immediately applicable to propagation of waves in single-mode optical waveguides. The optical methods are more efficient and provide possibility to measure phases ψ and χ. On the one hand, it makes observable the phase transition in the distribution P(ψ), which can be considered as a “trace” of the mobility edge remaining in 1D systems. On the other hand, observability of the phase χ makes actual derivation of its evolution equation, which is presented below. Relaxation of the distribution P(χ) to the limiting distribution P∞ (χ) at L → ∞ is described by two exponents, whose exponentials have jumps of the second derivative, when the energy Ɛ is changed.