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Quantum fidelity susceptibility in excited state quantum phase transitions: application to the bending spectra of nonrigid molecules
by J. KhaloufRivera, M. Carvajal, F. PérezBernal
Submission summary
As Contributors:  Jamil KhaloufRivera · Francisco PerezBernal 
Arxiv Link:  https://arxiv.org/abs/2102.12335v2 (pdf) 
Date submitted:  20210308 09:16 
Submitted by:  PerezBernal, Francisco 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We characterize excited state quantum phase transitions in the two dimensional limit of the vibron model with the quantum fidelity susceptibility, comparing the obtained results with the information provided by the participation ratio. As an application, we perform fits using a fourbody algebraic Hamiltonian to bending vibrational data for several molecular species and, using the optimized eigenvalues and eigenstates, we locate the eigenstate closest to the barrier to linearity and determine the linear or bent character of the different overtones.
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Anonymous Report 1 on 2021517 (Invited Report)
Report
The work “Quantum fidelity susceptibility in excited state quantum phase transitions: application to the bending spectra of nonrigid molecules” by KhaloufRivera, Carvajal and P erezBernal represents an extension of the analysis performed by the same authors in Ref.[33]. The purpose of these works is to identify (ES)QPTs and quantum phases in the spectrum of the twodimensional vibron model with a general Hamiltonian containing terms up to the fourbody interactions. As shown in the previous work, the model in this form provides a satisfactory description of experimental data on numerous molecules, so it is certainly worth of attention.
In the previous work, the main tool to distinguish quantum phases was the participation ratio, which measures the localization of eigenstates in a selected basis. However, this quantity fails to perform the desired task of an ESQPT indicator in generic systems in which ESQPTs do not show localization effects in the initial eigenbasis. Here the authors employ the socalled fidelity susceptibility (FS) of the eigenstates. This is just the lowestorder (quadratic) term of the fidelity expansion in powers of the local parameter change. Since the FS quantifies how rapid is the local change of the selected eigenstate with the control parameter, it can serve as an indicator of structural changes and ESQPTs in the spectrum.
The work is well written and presents interesting new results. The authors show the behavior of the FS not only in the l=0 eigenstates, that undergo (in the infinite size limit) the ESQPT, but also in the l>0 ones, which have no ESQPT and show only fading signatures of the structural change. The method based on the FS is proven to be a useful heuristic tool for fast localization of rapid structural changes of eigenstates in the model parameter space, including excitedstate critical effects. Applications of this type may follow also in other models, whose complexity hinders the full semiclassical analysis needed for the identification of ESQPTs.
I recommend the present work for publication in the SciPost Physics. Below I give a few minor comments which may help to improve the clarity of the paper.
Requested changes
Here are some reccommendations for the authors:
1) I believe that the authors should mention that the FS is supposed to be large not only in crossing the QPT and ESQPT critical points, but also near any sufficiently sharp avoided crossing of levels. This directly follows from Eq.(8). So particularly in chaotic systems the present approach may face a trouble.
2) Concerning the origin of the concept of quantum monodromy, it would be convenient to cite besides the Child’s work also Cushman and Duistermaat, Bulletin of Am. Math. Soc. 19, 1988.
3) On page 3 the authors say that “For a system with n effective degrees of freedom, the order of the derivative of the level density that is nonanalytic is n–1.” However, this is true only in some cases (though the most common ones), particularly for ESQPTs caused by nondegenerate stationary points. A short comment on the other cases would be helpful.
4) Equation (7) deserves more motivation. It would be worth to say that the first derivative of the fidelity vanishes, so that the FS is the leadingorder term in the fidelity expansion.
5) The introduction of two control parameters, namely xi and lambda, is somewhat confusing, especially in Eq.(12), where both parameters play essentially the same role. Maybe a few motivating words would be in order here.
Author: Francisco PerezBernal on 20210608
(in reply to Report 1 on 20210517)We are very grateful for the careful reading of the manuscript by the invited referee and for his very positive opinion on our work. As she/he has stated, the fidelity susceptibility is very sensitive to eigenstate changes and this fact makes this quantity a fine probe to locate eigenstates with respect to an ESQPT separatrix, even for finite system sizes and angular momentum values other than zero. We are currently planning to extend the present approach to other systems and models, exploring a novel and convenient way to assign excited states to ESQPT "phases."
We proceed to answer the referee's five recommended changes:
(1) We thank the referee for this insightful remark. We have changed the text under Eq. (11) in page 9 to explain this point and the sensitivity of the QFS to avoided energy crossings.
(2) The reference has been included.
(3) We thank the referee for this suggestion that corrects our statement. The sentence has been recast to exclude degenerate stationary point cases.
(4) We agree with the referee that the QFS needs to be properly introduced in the paper, especially for those nonfamiliar with this quantity. We have introduced a sentence stating that the QFS is the leading term in the series expansion of the fidelity.
(5) We have introduced an explanatory sentence and changed \xi to its numerical value to clarify the role of the control parameter \lambda.
The final version with the changes described above will be uploaded once we include possible changes suggested by the expected second referee report.