Theory of random multiplicative transfer matrices and its implications for quantum transport

Abstract

The two-probe conductance, g, of a disordered quantum system with N tranverse scattering channels is determined by N real parameters ("levels") {λi} characterizing the transfer matrix M. The appropriate measure for M combined with a "global" maximum entropy hypothesis, leads to a joint distribution for these levels of the same form as the standard random matrix ensembles, except for the occurence of a novel behavior of the level density ρ(λ), which is far from uniform, and depends importantly on the system parameters. We study this density and its implications for conductance fluctuations in detail, showing that the novel behavior of this ensemble stems from the multiplicative composition law for M. For fixed N, as the system length Lz → oo it is the αi ~(1/2Lz) ℓn λ i (which converge to the Lyapunov exponents of M) that have an approximately uniform density. Using this density we develop a Coulomb gas analogy to understand analytically the change in the level and conductance statistics which accompanies the transition from metallic to localized behavior. The metallic regime corresponds to the high-density "phase" of the gas, with statistics similar to standard, logarithmically-correlated ensembles except for the appearance of an "interaction" with image charges near the origin; this leads to a normal distribution p(g) with a universal variance. Possible mechanisms for the occurrence of lognormal tails in the metallic regime are discussed. The localized regime corresponds to a low-density "phase" in which the levels fluctuate independently and the conductance is lognormally distributed with ⟨(δℓn g)2⟩ ~ 2/g0 where g0 = N ℓ/Lz is the classical conductance and ℓ is the elastic man free path. The consequences of the Coulomb gas analogy for both the conductance and level statistics are confirmed by independent numerical calculations. In our theory, the distribution p(g) depending only on p(A), we study in a second part of this work the finite-size scaling properties of ρ(λ) to address the question of one-parameter scaling. First we generalize for each level α i the scaling law obtained for the disordered chain. Then we show that the convergence of each αi towards their quasi-one dimensional limit (N fixed, Lz → oo) depends only on the average conductance (g) in the range of investigated parameters. However, we cannot rule out a dependence on the index i of the scaling functions which would introduce additional scaling parameters for p(g)