Sigma models for quantum chaotic dynamics

TL;DR

This paper reviews the supersymmetric sigma model for quantum chaos, demonstrating how classical chaos influences spectral fluctuations and localization phenomena in different quantum systems.

Contribution

It introduces the construction of the supersymmetric sigma model for unitary maps and applies it to three case studies in quantum chaos.

Findings

Universal spectral fluctuations occur in fully chaotic systems.

Quantum localization arises due to long-lived diffusive modes.

The sigma model framework explains differences between chaotic and localized quantum systems.

Abstract

We review the construction of the supersymmetric sigma model for unitary maps, using the color- flavor transformation. We then illustrate applications by three case studies in quantum chaos. In two of these cases, general Floquet maps and quantum graphs, we show that universal spectral fluctuations arise provided the pertinent classical dynamics are fully chaotic (ergodic and with decay rates sufficiently gapped away from zero). In the third case, the kicked rotor, we show how the existence of arbitrarily long-lived modes of excitation (diffusion) precludes universal fluctuations and entails quantum localization.