Boundary Conditions, Phase Distribution, and Hidden Symmetry in 1D Localization

One-dimensional disordered systems with a random potential of a small amplitude and short-range correlations are considered near the initial band edge. The evolution equation is obtained for the mutual distribution P (ρ, ψ) of the Landauer resistance ρ and the phase variable ψ = θ – φ (θ and φ are phases entering the transfer matrix) when the system length L is increased. In the limit of large L , the equation allows separation of variables, which provides the existence of the stationary distribution P (ψ), determining the coefficients in the evolution equation for P (ρ). The limiting distribution P (ρ) for L → ∞ is log-normal and does not depend on boundary conditions. It is determined by the “internal” phase distribution, whose form is established in the whole energy range including the forbidden band of the initial crystal. The random phase approximation is valid in the depth of the allowed band, but strongly violated for other energies. The phase ψ appears to be the “bad” variable, while the “correct” variable is ω = –cotψ/2. The form of the stationary distribution P (ω) is determined by the internal properties of the system and is independent of boundary conditions. Variation of the boundary conditions leads to the scale transformation ω → s ω and translations ω → ω + ω 0 and ψ → ψ + ψ 0 , which determinates the “external” phase distribution entering the evolution equations. Independence of the limiting distribution P (ρ) on the external distribution P (ψ) makes it possible to speak about the hidden symmetry, whose character is revealed below.