Exact solution for the distribution of transmission eigenvalues in a disordered wire and comparison with random-matrix theory.

Abstract

We consider the complete probability distribution P({${\mathit{T}}_{\mathit{n}}$}) of the transmission eigenvalues ${\mathit{T}}_{1}$,${\mathit{T}}_{2}$,...,${\mathit{T}}_{\mathit{N}}$ of a disordered quasi-one-dimensional conductor (length L much greater than width W and mean free path l). The Fokker-Planck equation which describes the evolution of P with increasing L is mapped onto a Schr\"odinger equation by a Sutherland-type transformation. In the absence of time-reversal symmetry (e.g., because of a magnetic field), the mapping is onto a free-fermion problem, which we solve exactly. The resulting distribution is compared with the predictions of random-matrix theory (RMT) in the metallic regime (L\ensuremath{\ll}Nl) and in the insulating regime (L\ensuremath{\gg}Nl). We find that the logarithmic eigenvalue repulsion of RMT is exact for ${\mathit{T}}_{\mathit{n}}$'s close to unity, but overestimates the repulsion for weakly transmitting channels. The nonlogarithmic repulsion resolves several long-standing discrepancies between RMT and microscopic theory, notably in the magnitude of the universal conductance fluctuations in the metallic regime, and in the width of the log-normal conductance distribution in the insulating regime.