A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics

Abstract

We present a formal geometric framework for the study of adiabatic quantum mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states appearing for non-Hermitian Hamiltonians. We start with an investigation of the space of non-degenerate operators on a finite-dimensional state space. We then show how the energy bands of a Hamiltonian family form a covering space. Likewise, we show that the eigenrays form a bundle, a generalization of a principal bundle, which admits a natural connection yielding the (generalized) geometric phase. This bundle provides in addition a natural generalization of the quantum geometric tensor and derived tensors, and we show how it can incorporate the non-geometric dynamical phase as well. We finish by demonstrating how the bundle can be recast as a principal bundle, so that both the geometric phases and the permutations of eigenstates can be expressed simultaneously by means of standard holonomy theory.

On the Geometry of Adiabatic Quantum Mechanics

“…The space of non-degenerate operators Motivated by their prominence in the adiabatic approximation, let us now consider non-degenerate operators in more detail. This exploration will be the basis for all formal treatment to come, and was presented in [32]. We will not consider any specific operator family H(x), nor will we use any inner product on the vector space V .…”

“…Everything is then in place to use results from covering theory, and we find that the monodromy action yields a natural description for the energy swaps around EPs. This material has been published in [21,32].…”

“…This will bring us to the concept of a semi-principal bundle and G-connections, which extends the usual model of geometric phase to a model that incorporates non-cyclic states in a natural way. The material of this chapter has appeared in [32,42].…”

“…We will finish by discussing additional applications of the frame bundle, such as providing a geometric model for the off-diagonal phases found by Manini and Pistolesi [14]. Most of the material in this chapter has appeared in [32,42].…”

“…In our earlier work[32] we restricted to N Herm (V ), but the argument already works over N normal (V ).…”