Hellmann Feynman Theorem in Non-Hermitian system

TL;DR

This paper extends the Hellmann-Feynman theorem to PT-invariant non-Hermitian quantum systems, deriving a generalized version applicable across phases and at exceptional points, with potential experimental implications.

Contribution

It introduces a modified Hellmann-Feynman theorem for non-Hermitian systems, valid in PT-broken, unbroken phases, and at exceptional points, including non-PT-invariant Hamiltonians.

Findings

Modified HFT diverges at real-to-complex eigenvalue transition points.

Divergence also occurs at exceptional points in PT-invariant systems.

Derived a generalized Virial theorem for non-Hermitian quantum systems.

Abstract

We revisit the celebrated Hellmann-Feynman theorem (HFT) in the PT invariant non-Hermitian quantum physics framework. We derive a modified version of HFT by changing the definition of inner product and explicitly show that it holds good for both PT broken, unbroken phases and even at the exceptional point of the theory. The derivation is extremely general and works for even PT non-invariant Hamiltonian. We consider several examples of discrete and continuum systems to test our results. We find that if the eigenvalue goes through a real to complex transition as a function of the Hermiticity breaking parameter, both sides of the modified HFT expression diverge at that point. If that point turns out to be an EP of the PT invariant quantum theory, then one also sees the divergence at EP. Finally, we also derive a generalized Virial theorem for non-Hermitian systems using the modified HFT, which potentially can be tested in experiments.