Topology of quadrupolar Berry phase of a Qutrit

TL;DR

This paper explores the topological nature of the quadrupolar Berry phase in spin-1 systems, visualizing it via Majorana stars and analyzing its quantization through dynamics and symmetries.

Contribution

It introduces a geometric visualization of the quadrupolar Berry phase using Majorana representation and links its topological properties to symmetries and state evolution.

Findings

Quadrupolar Berry phase is topologically zero or pi.

Majorana stars tracing a great circle induce the pi phase.

Time evolution in the quadrupolar subspace produces quantized geometric phases.

Abstract

We examine Berry phase pertaining to purely quadrupolar state ($\langle \psi | \vec{S} | \psi \rangle = 0$) of a spin-$1$ system. Using the Majorana stellar representation of these states, we provide a visualization for the topological (zero or $\pi$) nature of such quadrupolar Berry phase. We demonstrates that the $\pi$ Berry phase of quadrupolar state is induced by the Majorana stars collectively tracing out a closed path (a great circle) by exchanging their respective positions on the Bloch sphere. We also analyse the problem from the perspective of dynamics where a state from the quadrupolar subspace is subjected to a static magnetic field. We show that time evolution generated by such Hamiltonian restricts the states to the quadrupolar subspace itself thereby producing a geometric phase (of the Aharonov-Anandan type) quantized to zero or $\pi$. A global unitary transformation which maps the quadrupolar subspace to the subspace of purely real states proves a natural way of understanding the topological character of this subspace and its connection to the anti-unitary symmetries.