Information Geometry in Time Dependent Quantum Systems and the Geometric Phase

TL;DR

This paper explores the geometric structure of time-dependent quantum systems, analyzing parameter manifolds, geometric phases, and their implications in models like the XY spin chain, revealing new insights and contradictions with prior studies.

Contribution

It provides a detailed geometric analysis of time-dependent quantum systems, including a critique of existing literature and proposing a potential new geometric phase across quantum critical lines.

Findings

Contradicts previous results on geometric phases in the XY model in the thermodynamic limit.

Identifies global properties of parameter manifolds in driven two-level systems.

Suggests a novel geometric phase when crossing quantum critical lines.

Abstract

We study information theoretic geometry in time dependent quantum mechanical systems. First, we discuss global properties of the parameter manifold for two level systems exemplified by i) Rabi oscillations and ii) quenching dynamics of the XY spin chain in a transverse magnetic field, when driven across anisotropic criticality. Next, we comment upon the nature of the geometric phase from classical holonomy analyses of such parameter manifolds. In the context of the transverse XY model in the thermodynamic limit, our results are in contradiction to those in the existing literature, and we argue why the issue deserves a more careful analysis. Finally, we speculate on a novel geometric phase in the model, when driven across a quantum critical line.