On the Geometry of Adiabatic Quantum Mechanics

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 non-Hermitian 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 non-trivial 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 non-degenerate 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 multi-state approach allowing one to express the results using explicit matrices.