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.
Highlights
If a system is in equilibrium with a heat bath at inverse temperature β, the state of the system is characterised by the canonical phase-space density function: ρ(x|β) =
For a real analytic function, it is not possible to find a breakdown of analyticity in a system having finitely many degrees of freedom, and it is mandatory to consider the operation of a thermodynamic limit
If the state of a system is described by a density function that lacks analyticity in the first place, a phase transition can be seen without involving the mathematically cumbersome operation of the thermodynamic limit
Summary
In the case of a complex Hamiltonian, the associated eigenfunctions need not be analytic in the parameters of the Hamiltonian, and phase transitions can be seen in finite matrix Hamiltonians (see, e.g., [2]) This situation is reminiscent of the analysis proposed by Lee and Yang [3,4], where the breakdown of analyticity associated with the canonical density function in Equation (1) can be explained by extending the parameters into a complex domain It is our hope that the present paper serves as a concise introduction to the physics of complex Hamiltonians for those who work in the area of information geometry and, at the same time, an introduction to information geometry for those who work in the study of physical systems described by complex Hamiltonians
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