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Physica A Statistical Mechanics and its Applications

Statistical properties of the Hamiltonian generating phase state derived by using the generalized Hellmann–Feynman theorem

Hong-Yi Fan, Xue-xiang Xu, Liyun Hu

Published Jan 5, 2010• 3 citations
Publisher
DOI
10.1016/j.physa.2009.12.054

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Physical Review A•Feb 20, 2024•12 citations
We revisit the celebrated Hellmann-Feynman theorem (HFT) in the parity-time (PT)(\mathcal{PT})(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\mathcal{PT}PT broken and unbroken phases and even at the exceptional point of the theory. The derivation is extremely general and works for even the PT\mathcal{PT}PT noninvariant Hamiltonian. We consider several examples of discrete and continuum variable 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\mathcal{PT}PT-invariant quantum theory, then one also sees the divergence at EP. Moreover, we have also demonstrated that using the modified HFT can give rise to a potential numerical advantage for computing the expectation value of many-body operators for interacting many-body Hamiltonian. Finally, we also derive a generalized virial theorem for non-Hermitian systems using the modified HFT, which potentially can be tested in experiments.