Exploring the Geometric and Dynamical Properties of Spin Systems and Their Interplay with Quantum Entanglement

TL;DR

This thesis investigates the geometric and dynamical aspects of quantum entanglement and spin systems, highlighting their interplay through phase space, Hilbert space geometry, and state evolution.

Contribution

It provides a comprehensive geometric and dynamical analysis of quantum states and spin systems, connecting classical phase space structures with quantum evolution and entanglement.

Findings

Classical phase space formalism parallels quantum Hilbert space structures.

Geometric interpretation of quantum evolution via Fubini-Study metric and phases.

Analysis of entanglement dynamics and evolution speed in spin models.

Abstract

This thesis, explores the quantum entanglement and evolution through both a geometric and dynamical perspective. The first part focuses on classical phase space and its central role in Hamiltonian mechanics, emphasizing the importance of symplectic structures in describing mechanical states. The study highlights the formal analogy between classical phase space and the Hilbert space used in quantum mechanics. The second part is devoted to the geometric description of quantum states through the projective structure of Hilbert space. Emphasis is placed on the geometric interpretation of quantum evolution, particularly via the Fubini-Study metric, associated symplectic structures, and the geometric phase acquired during unitary evolutions. The final two parts are dedicated to the study of spin systems (both two-body and many-body) under different interaction models (XXZ Heisenberg and all-range Ising). Both the dynamical aspects (evolution speed, entanglement, and the quantum brachistochrone problem) and the geometric and topological structures of the corresponding quantum states are analyzed.