Riemann surfaces of complex classical trajectories and tunnelling splitting in one-dimensional systems

TL;DR

This paper explores the topology of complex classical trajectories on Riemann surfaces to understand quantum tunnelling splittings in one-dimensional polynomial Hamiltonian systems, emphasizing the role of non-local effects and Stokes phenomena.

Contribution

It introduces a topological framework using Riemann surfaces to systematically identify all paths contributing to tunnelling splittings, revealing non-local quantization effects.

Findings

Complete enumeration of tunnelling paths via fundamental group analysis

Identification of action relations among disjoint classical regions

Highlighting the significance of Stokes phenomena in tunnelling calculations

Abstract

The topology of complex classical paths is investigated to discuss quantum tunnelling splittings in one-dimensional systems. Here the Hamiltonian is assumed to be given as polynomial functions, so the fundamental group for the Riemann surface provides complete information on the topology of complex paths, which allows us to enumerate all the possible candidates contributing to the semiclassical sum formula for tunnelling splittings. This naturally leads to action relations among classically disjoined regions, revealing entirely non-local nature in the quantization condition. The importance of the proper treatment of Stokes phenomena is also discussed in Hamiltonians in the normal form.