Exact WKB analysis for dynamical and geometric exponents in generalized and nonlinear Landau-Zener transitions

TL;DR

This paper employs exact WKB analysis to rigorously study dynamical and geometric exponents in generalized Landau-Zener models, revealing universal discontinuities linked to phase transitions and topological phenomena.

Contribution

It introduces the application of exact WKB to analyze non-perturbative geometric phases and exponents in Landau-Zener transitions, addressing limitations of conventional methods.

Findings

Discontinuity of complex geometric factor is universal.

Exact WKB provides rigorous analysis of non-perturbative phenomena.

Differences from other methods in deriving exponents are highlighted.

Abstract

The Berry phase is a geometric phase that is important in explaining topological quantum phenomena. The Berry phase is also important in non-perturbative phenomena, as the imaginary part of the phase explains the non-perturbative transitions. However, problems arose because the singular perturbation with respect to the Planck constant has not been treated adequately in conventional calculations, where the most serious problem is the arbitrariness of approximate calculations. To solve this problem, we consider the exact WKB, which is a mathematical method that treats perturbative expansion with respect to the Planck constant as a rigorous singular perturbation. This method is also a powerful computational tool that makes analytical computation much easier for mathematical software. Using the exact WKB, we analyze the derivation of the dynamical and the geometric exponents in generalized Landau-Zener models, highlighting the differences from other calculational methods. The discontinuity of complex geometric factor is a universal phenomenon that manifests itself in phase transitions, boundaries, particle generation, and topology changes. These phenomena are ``non-perturbative'' in physics, while mathematically, these discontinuities can be deeply related to the singular structure of complex analysis. The mathematical structure of these phenomena will be revealed by using the exact WKB.