Quantum geometric phase in Majorana's stellar representation: Mapping onto a many-body Aharonov-Bohm phase

TL;DR

This paper explores the geometric phase in quantum systems by mapping pure states onto a gas of Dirac strings and Majorana stars, revealing a many-body Aharonov-Bohm phase and providing new geometric insights into quantum dynamics.

Contribution

It introduces a novel geometric representation of quantum states using Majorana stars and Dirac strings, linking the geometric phase to a many-body Aharonov-Bohm effect.

Findings

Mapping of quantum states to Dirac string gas and Majorana stars.

Expressions for geometric connection, curvature, and multipole moments.

Application to systems with exotic orderings like spin nematics.

Abstract

The (Berry-Aharonov-Anandan) geometric phase acquired during a cyclic quantum evolution of finite-dimensional quantum systems is studied. It is shown that a pure quantum state in a (2J+1)-dimensional Hilbert space (or, equivalently, of a spin-J system) can be mapped onto the partition function of a gas of independent Dirac strings moving on a sphere and subject to the Coulomb repulsion of 2J fixed test charges (the Majorana stars) characterizing the quantum state. The geometric phase may be viewed as the Aharonov-Bohm phase acquired by the Majorana stars as they move through the gas of Dirac strings. Expressions for the geometric connection and curvature, for the metric tensor, as well as for the multipole moments (dipole, quadrupole, etc.), are given in terms of the Majorana stars. Finally, the geometric formulation of the quantum dynamics is presented and its application to systems with exotic ordering such as spin nematics is outlined.