Quantum fidelity susceptibility in excited state quantum phase transitions: application to the bending spectra of nonrigid molecules

We have already produced and uploaded replies to referees #1 and #2 comments . We enclose now a detailed list of the changes that we have introduced in the submitted work to accommodate the referees' suggestions, as well as a detailed list of minor changes.

However, as you already know from a previous communication, we would like to express our utter surprise by the transfer of our work from SciPost Physics to SciPost Physics Core. None of the referees suggested such a transfer. We understand that the overlap of this paper with a paper we recently published (Journal of Quantitative Spectroscopy & Radiative Transfer [261 (2021) 107436]), raised by referee #2, is what has motivated this decision. Let us delve further into this important issue.

As we explained in the response to #2 referee comments, the overlapping material was included to make the paper more self-contained and easier to read. In the JQSRT reference, we presented a 4-body algebraic 2D vibron model Hamiltonian and applied it to model bending vibrational spectra for a set of molecules. We also present results obtained using the inverse participation ratio to analyze the resulting eigenstates localization. However, the key point of the present paper, the extension to excited states of the Quantum Fidelity Susceptibility to characterize the different phases of an excited state quantum phase transition, is both completely original and not at all present in our previous paper. As an application to reported data, we analyze bending vibration data for six molecules using the 2D vibron model. For our calculations, in three of the cases we use wavefunctions and energy spectra obtained from optimized spectroscopic parameters published in JQSRT. But in the other three cases we have performed new fits, that are published in this case for the first time. Anyhow, we find that this is not substantial. Even if the six molecules were already treated in the JQSRT reference, the novelty in this work is the calculation of the quantum fidelity susceptibility and its application to ESQPTs. We hope that in the new version it is completely clear that the material included in this paper is completely original and worth of publication in SciPost Physics.

Changes introduced from Referee #1 suggestions:

#1 Page 2. We have introduced the Cushman and Duistermaat reference suggested by the referee. (Requested change #2)

#2 Page 3. We have replaced

"For a system with n effective degrees of freedom, the order of the derivative of the level density that is non-analytic is ..."

with

"For a system with a nondegenerate stationary point and \(n\) effective degrees of freedom, the order of the derivative of the level density that is non-analytic is ...". (Requested change #3)

#3 Page 7. We have replaced

"which is maximum when the parameter $\lambda$ goes through a critical value"

by

"which is the second order and leading- term in the series expansion of the fidelity as a function of \(\delta\lambda\) \cite{You2007,Gu2010}. This quantity is maximal when the parameter \(\lambda\) goes through a critical value". (Requested change #4)

#4 Page 8. We have replaced "A similar procedure has been recently published, using QFS of excited states, in the study of the adiabatic and counter-adiabatic driving in ESQPTs [67] and of the onset of quantum chaos in spin chain models [116]."

in the old version by

"The energy difference in the denominator of Eq. (9) makes QFS also very sensitive to the presence of avoided crossings, pervasive in chaotic systems. In fact, this quantity has been recently used to reveal the onset of quantum chaos in spin chain models [117]. In chaotic systems having ESQPTs, QFS is still a valid tool that ought to be combined with other quantities sensitive to the transition to chaos in the system. It is worth to mention that recently it has been published a study of adiabatic and counter-adiabatic driving in ESQPTs [68] using the extension of the QFS to excited states."

in the new one (Requested change #1).

#5 Page 8 We have completely rewritten the text around Eq. (10) in the new version to clarify the different roles of parameters xi and lambda. (Requested change #5)

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Changes introduced from Referee #2 suggestions:

#1 Abstract: We have rewritten it to make clear that we out of the six molecular species whose bending modes where studied, only the fits for three of them have been performed with this work in mind, the fit for the other three being recently published in the above mentioned JQSRT paper.

#2 Page 4. Last paragraph of the introductory section. we replaced

"In Sect. 4, we introduce the four-body Hamiltonian used to fit to experimental data and we apply the formalism to the Si2C molecule, a well-known example of nonrigid molecule [6]."

by

"In Sect. 4, we apply the formalism to the Si2C molecule, a well-known example of nonrigid molecule [6].".

#3 Page 5. We removed the branching rules between quantum labels in Chains I and II -old version Eq. (2)- replacing them by several references for the interested reader.

#4 Page 6. We removed Eq. (4) in the old version -model hamiltonian matrix elements in Chain I basis- and replace it by the reference to [29].

#5 Page 6. We removed Figure 1 and the text addressing this figure, as the correlation energy diagram for the model Hamiltonian can be found elsewhere and, though it may be enlightening to the reader nonfamiliar with the 2DVM, it can be found elsewhere.

#6 Page 6. We introduced at the end of section 1 the participation ratio as a tool to study wave function localization in ESQPTs, instead of making it in section 2 as in the original manuscript version. In this way, we concentrate in section 2 in the QFS which is the truly important contribution of the present work.

#7 Page 7. Section 3 is now entirely devoted to the QFS.

#8 Page 9. We removed the last sentence in the first paragraph of Section 4: "In particular, we show ... OCCCO." and we have recasted the text following Eq. (11) to make clear that the matrix elements of the four-body Hamiltonian can be found in the JQSRT reference and that for three of the addressed cases (Si2C, NCNCS, and HNC) we use eigenstates from the fits published in JQSRT, making clear that Table 1 only contains unpublished fit results.

#9 Page 13. We have removed any discussion of priorly published fit results, replacing them by the corresponding Ref. to the JQSRT article. E.g. First paragraph of Subsection 4.1 and second and third paragraphs of Subsection 4.2.

#10 Page 15, Table 1 in the new version only includes fit to bending degrees of freedom that have not been previously published (molecular species CH3NCO, 37ClCNO, and OCCCO. The references to the table in the text have been changed accordingly.

#11 Page 17. The beginning of the last section (Section 5) has been rewritten to emphasize that the main purpose of this manuscript is the use of QFS in the study of ESQPTs. The second paragraph has also been rewritten to make clear what is new in this article and what stems from the previously published JQSRT paper.

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Other (minor changes) that are mostly grammar mistakes and changes trying to better define the scope of our work:

#1 Page 2. "action angle" --> "action-angle"

#2 Reference [68] (new version) has been corrected and completed.

#3 Page 3 "link ESQPT and thermodynamic transitions" has been changed
to "a possible link between ESQPTs and thermodynamic transitions".

#4 Page 4 "using quantum fidelity or quantum fidelity susceptibility." has been changed to "using quantum fidelity susceptibility."

#5 Page 4 The sentences "In the present work we extend the calculation of quantum fidelity and QFS to 2DVM excited states. We obtain an unambiguous assignment of such states to one of the possible ESQPT phases, as we can locate the state position relative to the separatrix between ESQPT phases." have been replaced by "In the present work, we extend the calculation of QFS to 2DVM excited states, obtaining an unambiguous assignment of such states to one of the possible ESQPT phases, as we can locate the state position relative to the separatrix between ESQPT phases.".

#6 Page 4 The introduction last paragraph has been recasted to better define the scope of the different sections in the manuscript.

#7 Page 5 Last sentence. "first order" -> "first-order"

#8 Page 6 ",i t has been found that state(s) close to the critical energy of the ESQPT display a high localization in one of dyanamical symmetry basis" has been replaced by "it has been found that states close to the critical energy of the ESQPT display a high localization in a dynamical symmetry basis."

#9 Page 6 "This effect is blurred for increasing values of the vibrational angular momentum, $\ell$. This is an effect that can be explained by the centrifugal barrier precluding the wave function from exploring the barrier to linearity critical point." has been replaced by "This effect is blurred for increasing values of the vibrational angular momentum, $\ell$, as the centrifugal barrier precludes the wave function from exploring the barrier to linearity critical point.".

#10 Page 6 "where the $|[N]n = N^{\ell=0}\rangle$ component has the highest weight in the Chain (I) basis." has been replaced by "where the $|[N] n^\ell\rangle=|[N]N^{0}\rangle$ component has the highest weight in the Chain (I) basis.".

#11 Page 7. Prior to Eq. (4) in the new version we have added " and ground state \(\ket{\psi_0(\lambda)}\)".

#12 Page 7. Just after Eq. (4) we have replaced "ground quantum states" by "ground states".

#13 Page 8. At the end of the paragraph containing Eq. (6) we have
added "assuming a linear dependence of $\hat H(\lambda)$ in the
control parameter.".

#14 Page 8. Last paragraph. "The resulting diagram is, as expected, similar to the correlation energy of $\hat{\cal H}(\xi)$..." has been replaced by "The resulting diagram is, as expected, similar to the correlation energy diagram of the model Hamiltonian, $\hat{\cal H}(\xi)$...".

#15 Page 9. Third paragraph. We mention first QFS and, in second place, PR.

#16 Page 9. Fourth paragraph. We have replaced "black dashed" by "black-dashed" and write in the correct order "bent-like (linear)".

#17 Page 9. Last paragraph of section 3. We replaced "... fit of a Hamiltonian ..." with "... fit of spectroscopic parameters from an algebraic Hamiltonian ...".

#18 Page 13. After Eq. (12) we replaced "first order" with "first-order".

#19 Page 13. "As expected, in both cases, for the PR and the QFS, ESQPT precursors are weaker for higher $\ell$ values (see also Fig.~\ref{Si2C_lmb0} in Appendix \ref{AppA}). This is a well-known effect explained by ..." has been replaced by "QFS and PR for the ESQPT precursors are weaker for higher $\ell$ values, which is a well-known effect explained by ... "

#20 Page 16. "as two textbook examples of a linear molecule and of a nonrigid molecule." has been replaced with "as textbook examples of a linear and a nonrigid molecule, respectively.".

#21 Page 16. Last paragraph. Sentence "Therefore, their QFS maxima occur in the vicinity of $\lambda=0$." has been removed.

#22 Page 17. First paragraph. " ... would have bent character." has been replaced by " ... should be considered as a bent state.".