A topological formulation for exotic quantum holonomy

TL;DR

This paper introduces a topological framework to understand exotic quantum holonomy, describing how adiabatic cycles can interchange eigenstates and energies, with applications to two-level systems and crossing scenarios.

Contribution

It provides a topological formulation for eigenspace anholonomy, distinguishing large adiabatic cycles from small ones using homotopy theory, and explores both adiabatic and nonadiabatic cases.

Findings

Topological classification of adiabatic cycles affecting eigenspaces

Application to two-level quantum systems

Analysis of crossing and avoided crossing scenarios

Abstract

An adiabatic change of parameters along a closed path may interchange the (quasi-)eigenenergies and eigenspaces of a closed quantum system. Such discrepancies induced by adiabatic cycles are refereed to as the exotic quantum holonomy, which is an extension of the geometric phase. "Small" adiabatic cycles induce no change on eigenspaces, whereas some "large" adiabatic cycles interchange eigenspaces. We explain the topological formulation for the eigenspace anholonomy, where the homotopy equivalence precisely distinguishes the larger cycles from smaller ones. An application to two level systems is explained. We also examine the cycles that involve the adiabatic evolution across an exact crossing, and the diabatic evolution across an avoided crossing. The latter is a nonadiabatic example of the exotic quantum holonomy.