Abstract
The adiabatic theorem states that if the Hamiltonian of a quantum system is changed sufficiently slowly, then its instantaneous eigenstates are preserved. In this context, if the original Hamiltonian is restored at the end of the experiment, the phase that the eigenstate has acquired has a purely geometric contribution. This geometric, or Berry, phase is mathematically described using the theory of parallel transport. A generalization of the geometric phase for nonHermitian Hamiltonians was found by Garrison & Wright, but this does not fit in the standard parallel transport description. The main problem comes from special degeneracies of the Hamiltonian called exceptional points (EPs), around which the energy bands form spiral staircases and so have a nontrivial topology.
In this thesis, we introduce a mathematical framework that can cope with EPs and the generalized geometric phase simultaneously. We first treat how Hamiltonians can be studied without using an inner product. This works for all nondegenerate Hamiltonians, which form the first relevant space we study. We then find that the energy bands naturally form a covering space and conclude that the energy permutations due to EPs are described by the monodromy action. Afterwards, we extend this approach to eigenstates, where we find that the geometric phase comes from a natural connection. This provides a rigorous footing for topological geometric phase and the quantum geometric tensor. We finish by demonstrating an equivalent multistate approach allowing one to express the results using explicit matrices.
In this thesis, we introduce a mathematical framework that can cope with EPs and the generalized geometric phase simultaneously. We first treat how Hamiltonians can be studied without using an inner product. This works for all nondegenerate Hamiltonians, which form the first relevant space we study. We then find that the energy bands naturally form a covering space and conclude that the energy permutations due to EPs are described by the monodromy action. Afterwards, we extend this approach to eigenstates, where we find that the geometric phase comes from a natural connection. This provides a rigorous footing for topological geometric phase and the quantum geometric tensor. We finish by demonstrating an equivalent multistate approach allowing one to express the results using explicit matrices.
Original language  English 

Qualification  Doctor of Philosophy 
Awarding Institution 

Supervisors/Advisors 

Award date  25Oct2022 
Place of Publication  [Groningen] 
Publisher  
Print ISBNs  9789464219036 
DOIs  
Publication status  Published  2022 