Spectral statistics in disordered metals: a trajectories approach

TL;DR

This paper introduces a semiclassical trajectory-based approach to analyze spectral correlations in disordered metals, simplifying calculations and extending the diagonal approximation to include nearly identical but non-identical paths.

Contribution

It extends the standard diagonal approximation to include self-intersecting trajectories, reducing diagram complexity and applying a semiclassical approach to spectral statistics in disordered conductors.

Findings

Calculated the two-level correlation function to 3-loop order.

Verified a one-parameter scaling hypothesis in 2D.

Proposed application of the weak diagonal approximation to chaotic systems.

Abstract

We show that the perturbative expansion of the two-level correlation function, $R(\omega)$, in disordered conductors can be understood semiclassically in terms of self-intersecting particle trajectories. This requires the extension of the standard diagonal approximation to include pairs of paths which are non-identical but have almost identical action. The number of diagrams thus produced is much smaller than in a standard field-theoretical approach. We show that such a simplification occurs because $R(\omega)$ has a natural representation as the second derivative of free energy $F(\omega)$. We calculate $R(\omega)$ to 3-loop order, and verify a one-parameter scaling hypothesis for it in 2d. We discuss the possibility of applying our ``weak diagonal approximation'' to generic chaotic systems.