Maximum entropy ansatz for transmission in quantum conductors: a quantitative study in two and three dimensions

The transmission of electrons through a disordered conductor is described by a transmission matrix $t$. In random matrix theory the joint probability distribution of the eigenvalues of $t^\dagger t$ can be derived from a maximum entropy ansatz in which the mean eigenvalue density is given as a constraint. For a microscopic Anderson model, we examine the density for different shapes of the conductor (quasi–1d,2d, 3d). For the high transmission modes the form of the density is independent of disorder, size and dimensionality. We derive expressions for the eigenvalue correlations implied by the maximum entropy ansatz and compare these with the actual correlations of the Anderson model spectrum. We find that the correlations are qualitatively correct in all dimensions. However, the ansatz does not reproduce the weak system shape dependence of the universal conductance fluctuations (UCF), giving always results close to the quasi- 1d UCF- value. A careful study of the variances of different appropriate quantities indicates that the ansatz is quantitatively exact in quasi-1d over the whole spectrum of $t.t^\dagger$, but correctly describes the correlations in higher dimensions on intervals which are larger in the bulk of the spectrum than near the edge. We show that eigenvalues near the edge of the spectrum, corresponding to high transmission or reflection, remain correlated to associated non isotopically distributed eigenvectors