Conductance in diffusive quasi-one-dimensional periodic waveguides: a semiclassical and random matrix study

We study quantum transport properties of finite periodic quasi-one-dimensional waveguides whose classical dynamics is diffusive. The system we consider is a scattering configuration, composed of a finite periodic chain of $L$ identical (classically chaotic and finite-horizon) unit cells, which is connected to semi-infinite plane leads at its extremes. Particles inside the cavity are free and only interact with the boundaries through elastic collisions; this means waves are described by the Helmholtz equation with Dirichlet boundary conditions on the waveguide walls. The equivalent to the disorder ensemble is an energy ensemble, defined over a classically small range but many mean level spacings wide. The number of propagative channels in the leads is $N$. We have studied the (adimensional) Landauer conductance $g$ as a function of $L$ and $N$ in the cosine-shaped waveguide and by means of our RMT periodic chain model. We have found that $<g(L)>$ exhibit two regimes. First, for chains of length $L\lesssim\sqrt{N}$ the dynamics is diffusive just like in the disordered wire in the metallic regime, where the typic ohmic scaling is observed with $<g(L)> = N/(L+1)$. In this regime, the conductance distribution is a Gaussian with small variance but which grows linearly with $L$. Then, in longer systems with $L\gg\sqrt{N}$, the periodic nature becomes relevant and the conductance reaches a constant asymptotic value $<g(L\to\infty)> \sim <N_B>$. The variance approaches a constant value $\sim\sqrt{N}$ as $L\to\infty$. Comparing the conductance using the unitary and orthogonal circular ensembles we observed that a weak localization effect is present in the two regimes.