Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2107.10674

Comment on "Eigenstate clustering around exceptional points"

Francisco Marcelo Fernández

Published Jul 21, 2021• 0 citations
arXivHTMLPDF

Abstract

We show that the author of a recent paper [arXiv:2008.04929] put forward some false statements about the eigenstates of Hermitian and non-Hermitian systems. We conjecture that one of the non-Hermitian Hamiltonians for a one-dimensional lattice is similar to an Hermitian one and, consequently, exhibits real eigenvalues. Present theoretical analysis of the eigenvalue equation suggests that one of the sets of numerical results in the criticized paper may not be correct.

DOI
10.48550/arxiv.2107.10674

Related papers

Reply to "Comment on `Winding around non-Hermitian singularities' by Zhong et al., Nat. Commun. 9, 4808 (2018)"

Qi Zhong, Mercedeh Khajavikhan, Demetrios N. Christodoulides +1 more

A comment has been recently posted on the arXiv (arXiv:1902.07504) that discuss our recent work on encircling multiple exceptional points in Nat. Commun. 9, 4808 (2018). In that comment, the authors claim that our approach is prone to errors. In discussing their findings, they also indicate that a method presented by their team in Phys. Rev. A 98, 023818 (2018) (presumed to give correct results in all situations) was published prior to our work. First, we would like to note that both their work and ours were posted on the arXiv within a week from each other. Second and more importantly, as we will show below, their analysis and conclusion concerning the validity of our approach as presented in the comment article (arXiv:1902.07504) are not correct. As we will demonstrate, a proper application of our method does indeed provide the correct results.

arXiv:1903.07580•Mar 18, 2019•0 citations

Comment on 'Winding around non-Hermitian singularities' by Zhong et al., Nat. Commun. 9, 4808 (2018)

Eric J. Pap, Daniël Boer, Holger Waalkens

In a recent paper entitled "Winding around non-Hermitian singularities" by Zhong et al., published in Nat. Commun. 9, 4808 (2018), a formalism is proposed for calculating the permutations of eigenstates that arise upon encircling (multiple) exceptional points (EPs) in the complex parameter plane of an analytic non-Hermitian Hamiltonian. The authors suggest that upon encircling EPs one should track the eigenvalue branch cuts that are traversed, and multiply the associated permutation matrices accordingly. In this comment we point out a serious shortcoming of this approach, illustrated by an explicit example that yields the wrong result for a specific loop. A more general method that has been published earlier by us and that does not suffer from this problem, is based on using fundamental loops. We briefly explain the method and list its various advantages. In addition, we argue that this method can be verified in a three wave-guide system, which then also unambiguously establishes the noncommutativity associated with encircling multiple EPs.

arXiv:1902.07504•Feb 20, 2019•0 citations

Hermitian and pseudo-Hermitian Hamiltonians of SU(1,1) system—Spectrum, exceptional point, quantum–classical correspondence

Ni Liu, Meng Luo, Z. D. Wang +1 more

We in this paper study the relation of hermiticity and energy spectrum for Hamiltonians consisting of $SU(1,1)$ generators. In contrast with the common belief, the transition from real to imaginary spectra can appear at an exceptional point for the Hermitian Hamiltonian of a conservative system. The imaginary domain of spectrum resulted from an inverted potential well unbounded from below. An outward force applies on the particle, which moves acceleratingly away from the central point similar to the minisuperspace model of expanding universe. The Hamiltonian remains Hermitian beyond the exceptional point in a price that the boson-operator realization of $SU(1,1)$ generator $\widehat{S}_{z}$ becomes non-Hermitian. Oppositely a non-Hermitian Hamiltonian (called pseudo-Hermitian) possesses real eigenvalues in the entire region of interaction constant, which increases the gradient of potential well. While the Hermitian interaction decreases the gradient continuously to zero, namely the exceptional point, where eigenstates are degenerate. We extend the non-Hermitian Hamiltonian to the periodically time-dependent and two-dimensional system. The probability density of wave function coincides with classical orbits, which are derived from the corresponding classical-counterpart of the non-Hermitian quantum Hamiltonian.

Results in Physics•Dec 20, 2023•2 citations

Comment on ‘Numerical estimates of the spectrum for anharmonic PT symmetric potentials’

Paolo Amore, Francisco Marcelo Fernández

We show that the authors of the commented paper draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In one of the studied examples the authors missed the real positive eigenvalues that already converge towards the exact eigenvalues of the non-Hermitian operator and focused their attention on the complex ones that do not. We also show that the authors misread Bender's argument about the eigenvalues of the harmonic oscillator with boundary conditions in the complex-$x$ plane (Rep. Prog. Phys. {\bf 70} (2007) 947).

arXiv:1209.6357•Physica Scripta•Mar 8, 2013•1 citation

Imaginary Eigenvalues of Hermitian Hamiltonian with an Inverted Potential Well and Transition to the Real Spectrum at Exceptional Point by a Non-Hermitian Interaction

Ni Liu, Meng Luo, J.-Q. Liang +1 more

•Jan 1, 2024•0 citations

Imaginary Eigenvalues of Hermitian Hamiltonian with an Inverted Potential Well and Transition to the Real Spectrum at Exceptional Point by a Non-Hermitian Interaction

Ni Liu, Meng Luo, J.-Q. Liang

•Jan 1, 2024•0 citations
We show that the author of a recent paper [arXiv:2008.04929] put forward some false statements about the eigenstates of Hermitian and non-Hermitian systems. We conjecture that one of the non-Hermitian Hamiltonians for a one-dimensional lattice is similar to an Hermitian one and, consequently, exhibits real eigenvalues. Present theoretical analysis of the eigenvalue equation suggests that one of the sets of numerical results in the criticized paper may not be correct.
A comment has been recently posted on the arXiv (arXiv:1902.07504) that discuss our recent work on encircling multiple exceptional points in Nat. Commun. 9, 4808 (2018). In that comment, the authors claim that our approach is prone to errors. In discussing their findings, they also indicate that a method presented by their team in Phys. Rev. A 98, 023818 (2018) (presumed to give correct results in all situations) was published prior to our work. First, we would like to note that both their work and ours were posted on the arXiv within a week from each other. Second and more importantly, as we will show below, their analysis and conclusion concerning the validity of our approach as presented in the comment article (arXiv:1902.07504) are not correct. As we will demonstrate, a proper application of our method does indeed provide the correct results.
In a recent paper entitled "Winding around non-Hermitian singularities" by Zhong et al., published in Nat. Commun. 9, 4808 (2018), a formalism is proposed for calculating the permutations of eigenstates that arise upon encircling (multiple) exceptional points (EPs) in the complex parameter plane of an analytic non-Hermitian Hamiltonian. The authors suggest that upon encircling EPs one should track the eigenvalue branch cuts that are traversed, and multiply the associated permutation matrices accordingly. In this comment we point out a serious shortcoming of this approach, illustrated by an explicit example that yields the wrong result for a specific loop. A more general method that has been published earlier by us and that does not suffer from this problem, is based on using fundamental loops. We briefly explain the method and list its various advantages. In addition, we argue that this method can be verified in a three wave-guide system, which then also unambiguously establishes the noncommutativity associated with encircling multiple EPs.
We in this paper study the relation of hermiticity and energy spectrum for Hamiltonians consisting of SU(1,1)SU(1,1)SU(1,1) generators. In contrast with the common belief, the transition from real to imaginary spectra can appear at an exceptional point for the Hermitian Hamiltonian of a conservative system. The imaginary domain of spectrum resulted from an inverted potential well unbounded from below. An outward force applies on the particle, which moves acceleratingly away from the central point similar to the minisuperspace model of expanding universe. The Hamiltonian remains Hermitian beyond the exceptional point in a price that the boson-operator realization of SU(1,1)SU(1,1)SU(1,1) generator S^z\widehat{S}_{z}Sz​ becomes non-Hermitian. Oppositely a non-Hermitian Hamiltonian (called pseudo-Hermitian) possesses real eigenvalues in the entire region of interaction constant, which increases the gradient of potential well. While the Hermitian interaction decreases the gradient continuously to zero, namely the exceptional point, where eigenstates are degenerate. We extend the non-Hermitian Hamiltonian to the periodically time-dependent and two-dimensional system. The probability density of wave function coincides with classical orbits, which are derived from the corresponding classical-counterpart of the non-Hermitian quantum Hamiltonian.
We show that the authors of the commented paper draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In one of the studied examples the authors missed the real positive eigenvalues that already converge towards the exact eigenvalues of the non-Hermitian operator and focused their attention on the complex ones that do not. We also show that the authors misread Bender's argument about the eigenvalues of the harmonic oscillator with boundary conditions in the complex-xxx plane (Rep. Prog. Phys. {\bf 70} (2007) 947).