Information Geometry of Complex Hamiltonians and Exceptional Points

TL;DR

This paper explores the geometric structure of parameter spaces in complex non-Hermitian Hamiltonians, focusing on phase transitions and exceptional points, using information geometry and Fisher-Rao metrics to analyze sensitivity.

Contribution

It introduces a general geometric framework for analyzing eigenstates of complex Hamiltonians, providing explicit expressions for the Fisher-Rao metric near phase transitions.

Findings

Identifies high parameter sensitivity near phase transition points

Provides a systematic geometric approach to study exceptional points

Derives generic formulas for the metric tensor in complex Hamiltonian systems

Abstract

Information geometry provides a tool to systematically investigate parameter sensitivity of the state of a system. If a physical system is described by a linear combination of eigenstates of a complex (that is, non-Hermitian) Hamiltonian, then there can be phase transitions where dynamical properties of the system change abruptly. In the vicinities of the transition points, the state of the system becomes highly sensitive to the changes of the parameters in the Hamiltonian. The parameter sensitivity can then be measured in terms of the Fisher-Rao metric and the associated curvature of the parameter-space manifold. A general scheme for the geometric study of parameter-space manifolds of eigenstates of complex Hamiltonians is outlined here, leading to generic expressions for the metric.