Periodic instantons

Periodic instantons are finite energy solutions in quantum field theory that facilitate tunneling between two points in a potential barrier, also referred to as bounces. Vacuum instantons, which correspond to zero energy configurations in the limit of infinite Euclidean time, carry a topological winding number, unlike other configurations such as sphalerons—field states at the peak of a potential barrier.

Periodic instantons were first identified through explicit solutions of Euclidean-time field equations for double-well and cosine potentials, expressed using Jacobian elliptic functions. These solutions describe oscillations between two potential wells, with the frequency \( \Omega \) related to the energy difference \( \Delta E \) across the barrier by \( \Omega = \Delta E / \hbar \). The evaluation of \( \Delta E \) via path integrals involves summing over an infinite number of periodic instanton pairs, a process conducted in the dilute gas approximation.

These concepts have been applied across various theoretical domains, including quantum mechanics with anharmonic potentials, macroscopic spin systems (notably in studies by Garanin and Chudnovsky), the two-dimensional abelian Higgs model, four-dimensional electro-weak theories, and Bose-Einstein condensation scenarios involving tunneling between double-well traps.