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

Tobias Gulden, Michael Janas, Alex Kamenev

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2 Scopus citations


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.

Original languageEnglish (US)
Article number075304
JournalJournal of Physics A: Mathematical and Theoretical
Issue number7
StatePublished - Feb 20 2015


  • Algebraic topology
  • Riemann surface calculations
  • Semiclassical theories and applications
  • Single-molecule magnets

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