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Entropy 2013, 15(9), 3361-3378; doi:10.3390/e15093361

Information Geometry of Complex Hamiltonians and Exceptional Points

1
Mathematical Sciences, Brunel University, Uxbridge UB8 3PH, UK
2
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*
Author to whom correspondence should be addressed.
Received: 16 July 2013 / Revised: 12 August 2013 / Accepted: 16 August 2013 / Published: 23 August 2013
(This article belongs to the Special Issue Distance in Information and Statistical Physics Volume 2)
View Full-Text   |   Download PDF [293 KB, 24 February 2015; original version 24 February 2015]

Abstract

Information geometry provides a tool to systematically investigate the 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.
Keywords: information geometry; non-Hermitian Hamiltonian; perturbation theory; Fisher- Rao metric; phase transition; exceptional point; PT symmetry information geometry; non-Hermitian Hamiltonian; perturbation theory; Fisher- Rao metric; phase transition; exceptional point; PT symmetry
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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Brody, D.C.; Graefe, E.-M. Information Geometry of Complex Hamiltonians and Exceptional Points. Entropy 2013, 15, 3361-3378.

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