The semiclassical origin of the logarithmic singularity in the symplectic form factor

Sieber and Richter achieved a breakthrough towards a proof of the universality of spectral fluctuations of chaotic quantum systems conjectured by Bohigas, Giannoni and Schmidt by calculating semiclassically the first term beyond the diagonal approximation of the orthogonal form factor. In this letter, the semiclassical origin of the logarithmic singularity of the symplectic form factor is deduced perturbatively by combining this result with the contribution that arises due to the spin. This approach stands in contrast to the duality approach introduced by Bogomolny and Keating, which is essentially non-perturbative, and where the structure around the Heisenberg time is related to the structure for very small time which can be deduced using the diagonal approximation.