Quantum Geometric Tensor (Fubini-Study Metric) in Simple Quantum System: A pedagogical Introduction

TL;DR

This paper provides a pedagogical introduction to the Quantum Geometric Tensor, explaining its mathematical structure and physical significance in simple quantum systems, including adiabatic and non-adiabatic cases.

Contribution

It offers a clear, educational overview of the Quantum Geometric Tensor and its role in quantum geometry, bridging formalism with physical interpretation.

Findings

Clarifies the structure of the Quantum Geometric Tensor

Demonstrates its significance in adiabatic and non-adiabatic systems

Highlights the tensor's real and imaginary parts in physical phenomena

Abstract

Geometric Quantum Mechanics is a novel and prospecting approach motivated by the belief that our world is ultimately geometrical. At the heart of that is a quantity called Quantum Geometric Tensor (or Fubini-Study metric), which is a complex tensor with the real part serving as the Riemannian metric that measures the `quantum distance', and the imaginary part being the Berry curvature. Following a physical introduction of the basic formalism, we illustrate its physical significance in both the adiabatic and non-adiabatic systems.