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Information Geometry of Complex Hamiltonians and Exceptional Points
Mathematical Sciences, Brunel University, Uxbridge UB8 3PH, UK
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
* Author to whom correspondence should be addressed.
Received: 16 July 2013; in revised form: 12 August 2013 / Accepted: 16 August 2013 / Published: 23 August 2013
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
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Brody, D.C.; Graefe, E.-M. Information Geometry of Complex Hamiltonians and Exceptional Points. Entropy 2013, 15, 3361-3378.
Brody DC, Graefe E-M. Information Geometry of Complex Hamiltonians and Exceptional Points. Entropy. 2013; 15(9):3361-3378.
Brody, Dorje C.; Graefe, Eva-Maria. 2013. "Information Geometry of Complex Hamiltonians and Exceptional Points." Entropy 15, no. 9: 3361-3378.