Bions and Instantons in Triple-well and Multi-well Potentials

Dunne, Gerald V.; Sulejmanpasic, Tin; Ünsal, Mithat

Abstract

Quantum systems with multiple degenerate classical harmonic minima exhibit new non-perturbative phenomena which are not present for the double-well and periodic potentials. The simplest characteristic example of this family is the triple-well potential. Despite the fact that instantons are exact semiclassical solutions with finite and minimal action, they do not contribute to the energy spectrum at leading order in the semiclassical analysis. This is because the instanton fluctuation prefactor vanishes, which can be interpreted as the action becoming infinite quantum mechanically. Instead, the non-perturbative physics is governed by different types of bion configurations. A generalization to supersymmetric and quasi-exactly soluble models is also discussed. An interesting pattern of interference between topological and neutral bions, depending on the hidden topological angle, the discrete θ angle and the perturbative level number, leads to an intricate pattern of divergent/convergent expansions for low-lying states, and provides criteria for the exact solvability of some of the states. We confirm these semiclassical bion predictions using the BenderWu Mathematica package to study the structure of the associated perturbative expansions. It also turns out that all the systems we study have a curious exact one-to-one relationship between the perturbative coefficients of the three wells, which we check using the BenderWu package.

Citation

Dunne, G. V., Sulejmanpasic, T., & Ünsal, M. (2020). Bions and Instantons in Triple-well and Multi-well Potentials. In A. Niemi, T. Tomboulis, & K. K. Phua (Eds.), Roman Jackiw: 80th Birthday Festschrift (119-148). World Scientific Publishing. https://doi.org/10.1142/9789811210679_0015

• Online Publication Date: Aug 1, 2020
• Publication Date: 2020
• Deposit Date: Mar 13, 2020
• Publicly Available Date: Oct 28, 2022
• Publisher: World Scientific Publishing
• Pages: 119-148
• Book Title: Roman Jackiw: 80th Birthday Festschrift
• Chapter Number: 15
• DOI: https://doi.org/10.1142/9789811210679_0015
• Public URL: https://durham-repository.worktribe.com/output/1651955