Can the trace formula describe weak localisation?

TL;DR

This paper develops a semiclassical approach to derive quantum corrections to spectral correlations in chaotic systems, but finds that the method fails to accurately reproduce known weak localisation effects, especially at higher orders.

Contribution

It introduces a weak diagonal approximation based on the Gutzwiller trace formula to compute weak localisation corrections, highlighting its limitations.

Findings

Fails to reproduce Hikami boxes correctly

Incorrect first order correction prefactor

Inability to match standard second order results

Abstract

We attempt to systematically derive perturbative quantum corrections to the Berry diagonal approximation of the two-level correlation function (TLCF) for chaotic systems. To this end, we develop a ``weak diagonal approximation'' based on a recent description of the first weak localisation correction to conductance in terms of the Gutzwiller trace formula. This semiclassical method is tested by using it to derive the weak localisation corrections to the TLCF for a semiclassically disordered system. Unfortunately the method is unable to correctly reproduce the ``Hikami boxes'' (the relatively small regions where classical paths are glued together by quantum processes). This results in the method failing to reproduce the well known weak localisation expansion. It so happens that for the first order correction it merely produces the wrong prefactor. However for the second order correction, it is unable to reproduce certain contributions, and leads to a result which is of a different form to the standard one.