Instanton calculus without equations of motion: semiclassics from monodromies of a Riemann surface

TL;DR

This paper introduces a novel semiclassical instanton calculus that avoids solving classical equations of motion by leveraging monodromy properties of Riemann surfaces, simplifying calculations in complex quantum systems like single molecule magnets.

Contribution

It develops a method to compute instanton actions using Riemann surface monodromies, bypassing the need for explicit classical trajectory solutions.

Findings

Actions can be derived from monodromy properties and residues.

The method applies to spin-coherent states in complex phase spaces.

It successfully analyzes tunneling quenching in single molecule magnets.

Abstract

Instanton calculations in semiclassical quantum mechanics rely on integration along trajectories which solve classical equations of motion. However in systems with higher dimensionality or complexified phase space these are rarely attainable. A prime example are spin-coherent states which are used e.g. to describe single molecule magnets (SMM). We use this example to develop instanton calculus which does not rely on explicit solutions of the classical equations of motion. Energy conservation restricts the complex phase space to a Riemann surface of complex dimension one, allowing to deform integration paths according to Cauchy's integral theorem. As a result, the semiclassical actions can be evaluated without knowing actual classical paths. Furthermore we show that in many cases such actions may be solely derived from monodromy properties of the corresponding Riemann surface and residue values at its singular points. As an example, we consider quenching of tunneling processes in SMM by an applied magnetic field.