Chaotic quantum transport through spatially symmetric microstructures in the symplectic ensemble

Quantum transport through left-right symmetric chaotic cavities in the presence of the symplectic symmetry, is studied through the statistical distribution of the dimensionless conductance. With this particular point symmetry, their associated scattering matrices are blocky diagonalized by a rotation by an angle of $\pi/4$. Although the formulation is established for an arbitrary number channels N, we present explicit calculations for N=1 and N=2, the last one showing the weak anti-localization phenomenon due to the symplectic symmetry.